%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV148^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n184.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 300.07s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV148^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n184.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 08:13:46 CDT 2014
% % CPUTime  : 300.07 
% 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 0x156f5a8>, <kernel.Type object at 0x156fcf8>) 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), ((forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq0:(a->Prop)), (((and (forall (Xw0:a), (((R Xx) Xw0)->(Xq0 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((R Xu) Xv))->(Xq0 Xv))))->(Xq0 Xw)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw0:a), (((S Xx) Xw0)->(Xq0 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((S Xu) Xv))->(Xq0 Xv))))->(Xq0 Xw))))->(Xq Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq Xu)) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((R Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((S Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))->(Xq Xv))))->(Xq Xy)))->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xx) Xw)) ((S Xx) Xw))->(Xq Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq Xv))))->(Xq Xy))))) of role conjecture named cTHM251C_pme
% Conjecture to prove = (forall (R:(a->(a->Prop))) (S:(a->(a->Prop))) (Xx:a) (Xy:a), ((forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq0:(a->Prop)), (((and (forall (Xw0:a), (((R Xx) Xw0)->(Xq0 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((R Xu) Xv))->(Xq0 Xv))))->(Xq0 Xw)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw0:a), (((S Xx) Xw0)->(Xq0 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((S Xu) Xv))->(Xq0 Xv))))->(Xq0 Xw))))->(Xq Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq Xu)) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((R Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((S Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))->(Xq Xv))))->(Xq Xy)))->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xx) Xw)) ((S Xx) Xw))->(Xq Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq Xv))))->(Xq Xy))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (R:(a->(a->Prop))) (S:(a->(a->Prop))) (Xx:a) (Xy:a), ((forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq0:(a->Prop)), (((and (forall (Xw0:a), (((R Xx) Xw0)->(Xq0 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((R Xu) Xv))->(Xq0 Xv))))->(Xq0 Xw)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw0:a), (((S Xx) Xw0)->(Xq0 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((S Xu) Xv))->(Xq0 Xv))))->(Xq0 Xw))))->(Xq Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq Xu)) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((R Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((S Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))->(Xq Xv))))->(Xq Xy)))->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xx) Xw)) ((S Xx) Xw))->(Xq Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq Xv))))->(Xq Xy)))))']
% Parameter a:Type.
% Trying to prove (forall (R:(a->(a->Prop))) (S:(a->(a->Prop))) (Xx:a) (Xy:a), ((forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((or (forall (Xq0:(a->Prop)), (((and (forall (Xw0:a), (((R Xx) Xw0)->(Xq0 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((R Xu) Xv))->(Xq0 Xv))))->(Xq0 Xw)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw0:a), (((S Xx) Xw0)->(Xq0 Xw0)))) (forall (Xu:a) (Xv:a), (((and (Xq0 Xu)) ((S Xu) Xv))->(Xq0 Xv))))->(Xq0 Xw))))->(Xq Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq Xu)) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((R Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((S Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))->(Xq Xv))))->(Xq Xy)))->(forall (Xq:(a->Prop)), (((and (forall (Xw:a), (((or ((R Xx) Xw)) ((S Xx) Xw))->(Xq Xw)))) (forall (Xu:a) (Xv:a), (((and (Xq Xu)) ((or ((R Xu) Xv)) ((S Xu) Xv)))->(Xq Xv))))->(Xq Xy)))))
% Found x5:(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found x5 as proof of (Xq Xu0)
% Found x5:(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found x5 as proof of (Xq Xu0)
% Found x5:(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found x5 as proof of (Xq Xu0)
% Found x5:(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found x5 as proof of (Xq Xu0)
% Found x40:=(x4 x20):(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found (x4 x20) as proof of (Xq Xu0)
% Found (x4 x20) as proof of (Xq Xu0)
% Found x40:=(x4 x20):(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found (x4 x20) as proof of (Xq Xu0)
% Found (x4 x20) as proof of (Xq Xu0)
% Found or_intror00:=(or_intror0 ((S Xx) Xw0)):(((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (or_intror0 ((S Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (forall (Xw0:a), (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0))))
% Found or_introl00:=(or_introl0 ((S Xx) Xw0)):(((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (or_introl0 ((S Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (forall (Xw0:a), (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0))))
% Found x5:(Xq Xu)
% Found (fun (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5) as proof of (Xq Xu)
% Found (fun (x5:(Xq Xu)) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5) as proof of (((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->(Xq Xu))
% Found (fun (x5:(Xq Xu)) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5) as proof of ((Xq Xu)->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->(Xq Xu)))
% Found (and_rect10 (fun (x5:(Xq Xu)) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5)) as proof of (Xq Xu)
% Found ((and_rect1 (Xq Xu)) (fun (x5:(Xq Xu)) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5)) as proof of (Xq Xu)
% Found (((fun (P:Type) (x5:((Xq Xu)->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->P)))=> (((((and_rect (Xq Xu)) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))) P) x5) x4)) (Xq Xu)) (fun (x5:(Xq Xu)) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5)) as proof of (Xq Xu)
% Found (fun (x4:((and (Xq Xu)) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))))=> (((fun (P:Type) (x5:((Xq Xu)->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->P)))=> (((((and_rect (Xq Xu)) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))) P) x5) x4)) (Xq Xu)) (fun (x5:(Xq Xu)) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5))) as proof of (Xq Xu)
% Found (fun (Xv:a) (x4:((and (Xq Xu)) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))))=> (((fun (P:Type) (x5:((Xq Xu)->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->P)))=> (((((and_rect (Xq Xu)) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))) P) x5) x4)) (Xq Xu)) (fun (x5:(Xq Xu)) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5))) as proof of (((and (Xq Xu)) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))->(Xq Xu))
% Found (fun (Xu0:a) (Xv:a) (x4:((and (Xq Xu)) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))))=> (((fun (P:Type) (x5:((Xq Xu)->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->P)))=> (((((and_rect (Xq Xu)) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))) P) x5) x4)) (Xq Xu)) (fun (x5:(Xq Xu)) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5))) as proof of (forall (Xv:a), (((and (Xq Xu)) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))->(Xq Xu)))
% Found (fun (Xu0:a) (Xv:a) (x4:((and (Xq Xu)) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))))=> (((fun (P:Type) (x5:((Xq Xu)->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->P)))=> (((((and_rect (Xq Xu)) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))) P) x5) x4)) (Xq Xu)) (fun (x5:(Xq Xu)) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5))) as proof of (forall (Xu0:a) (Xv:a), (((and (Xq Xu)) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))->(Xq Xu)))
% Found x5:(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found (fun (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5) as proof of (Xq Xu0)
% Found (fun (x5:(Xq Xu)) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5) as proof of (((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->(Xq Xu0))
% Found (fun (x5:(Xq Xu)) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5) as proof of ((Xq Xu)->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->(Xq Xu0)))
% Found (and_rect10 (fun (x5:(Xq Xu)) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5)) as proof of (Xq Xu0)
% Found ((and_rect1 (Xq Xu0)) (fun (x5:(Xq Xu)) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5)) as proof of (Xq Xu0)
% Found (((fun (P:Type) (x5:((Xq Xu)->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((R Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((S Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->P)))=> (((((and_rect (Xq Xu)) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((R Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((S Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))) P) x5) x4)) (Xq Xu0)) (fun (x5:(Xq Xu)) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5)) as proof of (Xq Xu0)
% Found (((fun (P:Type) (x5:((Xq Xu)->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((R Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((S Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->P)))=> (((((and_rect (Xq Xu)) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((R Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((S Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))) P) x5) x4)) (Xq Xu0)) (fun (x5:(Xq Xu)) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5)) as proof of (Xq Xu0)
% Found x5:(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found (fun (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5) as proof of (Xq Xu0)
% Found (fun (x5:(Xq Xu)) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5) as proof of (((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->(Xq Xu0))
% Found (fun (x5:(Xq Xu)) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5) as proof of ((Xq Xu)->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->(Xq Xu0)))
% Found (and_rect10 (fun (x5:(Xq Xu)) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5)) as proof of (Xq Xu0)
% Found ((and_rect1 (Xq Xu0)) (fun (x5:(Xq Xu)) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5)) as proof of (Xq Xu0)
% Found (((fun (P:Type) (x5:((Xq Xu)->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((R Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((S Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->P)))=> (((((and_rect (Xq Xu)) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((R Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((S Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))) P) x5) x4)) (Xq Xu0)) (fun (x5:(Xq Xu)) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5)) as proof of (Xq Xu0)
% Found (((fun (P:Type) (x5:((Xq Xu)->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((R Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((S Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->P)))=> (((((and_rect (Xq Xu)) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((R Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((S Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))) P) x5) x4)) (Xq Xu0)) (fun (x5:(Xq Xu)) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5)) as proof of (Xq Xu0)
% Found x300:=(x30 x20):(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found (x30 x20) as proof of (Xq Xu0)
% Found ((x3 x10) x20) as proof of (Xq Xu0)
% Found ((x3 x10) x20) as proof of (Xq Xu0)
% Found x3:(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found x3 as proof of (Xq Xu0)
% Found x5:(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found x5 as proof of (Xq Xu0)
% Found x7:(Xq Xu)
% Found x7 as proof of (Xq Xu)
% Found x300:=(x30 x20):(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found (x30 x20) as proof of (Xq Xu0)
% Found ((x3 x10) x20) as proof of (Xq Xu0)
% Found ((x3 x10) x20) as proof of (Xq Xu0)
% Found or_intror00:=(or_intror0 ((S Xx) Xw0)):(((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (or_intror0 ((S Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (forall (Xw0:a), (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0))))
% Found or_introl00:=(or_introl0 ((S Xx) Xw0)):(((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (or_introl0 ((S Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (forall (Xw0:a), (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0))))
% Found x5:(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found x5 as proof of (Xq Xu0)
% Found or_intror00:=(or_intror0 ((S Xx) Xw0)):(((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (or_intror0 ((S Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (forall (Xw0:a), (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0))))
% Found or_introl00:=(or_introl0 ((S Xx) Xw0)):(((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (or_introl0 ((S Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (forall (Xw0:a), (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0))))
% Found x20:=(x2 x00):(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found (x2 x00) as proof of (Xq Xu0)
% Found (x2 x00) as proof of (Xq Xu0)
% Found x40:=(x4 x00):(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found (x4 x00) as proof of (Xq Xu0)
% Found (x4 x00) as proof of (Xq Xu0)
% Found or_introl00:=(or_introl0 ((S Xx) Xw0)):(((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (or_introl0 ((S Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (forall (Xw0:a), (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0))))
% Found or_intror00:=(or_intror0 ((S Xx) Xw0)):(((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (or_intror0 ((S Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (forall (Xw0:a), (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0))))
% Found or_introl00:=(or_introl0 ((S Xx) Xw0)):(((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (or_introl0 ((S Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (forall (Xw0:a), (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0))))
% Found or_intror00:=(or_intror0 ((S Xx) Xw0)):(((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (or_intror0 ((S Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (forall (Xw0:a), (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0))))
% Found x40:=(x4 x20):(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found (x4 x20) as proof of (Xq Xu0)
% Found (fun (x5:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> (x4 x20)) as proof of (Xq Xu0)
% Found (fun (x4:((forall (Xu00:a) (Xv:a), (((and (Xq Xu00)) ((or ((R Xu00) Xv)) ((S Xu00) Xv)))->(Xq Xv)))->(Xq Xu))) (x5:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> (x4 x20)) as proof of (((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->(Xq Xu0))
% Found (fun (x4:((forall (Xu00:a) (Xv:a), (((and (Xq Xu00)) ((or ((R Xu00) Xv)) ((S Xu00) Xv)))->(Xq Xv)))->(Xq Xu))) (x5:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> (x4 x20)) as proof of (((forall (Xu00:a) (Xv:a), (((and (Xq Xu00)) ((or ((R Xu00) Xv)) ((S Xu00) Xv)))->(Xq Xv)))->(Xq Xu))->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->(Xq Xu0)))
% Found (and_rect10 (fun (x4:((forall (Xu00:a) (Xv:a), (((and (Xq Xu00)) ((or ((R Xu00) Xv)) ((S Xu00) Xv)))->(Xq Xv)))->(Xq Xu))) (x5:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> (x4 x20))) as proof of (Xq Xu0)
% Found ((and_rect1 (Xq Xu0)) (fun (x4:((forall (Xu00:a) (Xv:a), (((and (Xq Xu00)) ((or ((R Xu00) Xv)) ((S Xu00) Xv)))->(Xq Xv)))->(Xq Xu))) (x5:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> (x4 x20))) as proof of (Xq Xu0)
% Found (((fun (P:Type) (x4:(((forall (Xu0:a) (Xv:a), (((and (Xq Xu0)) ((or ((R Xu0) Xv)) ((S Xu0) Xv)))->(Xq Xv)))->(Xq Xu))->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((R Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((S Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->P)))=> (((((and_rect ((forall (Xu0:a) (Xv0:a), (((and (Xq Xu0)) ((or ((R Xu0) Xv0)) ((S Xu0) Xv0)))->(Xq Xv0)))->(Xq Xu))) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((R Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((S Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))) P) x4) x3)) (Xq Xu0)) (fun (x4:((forall (Xu00:a) (Xv0:a), (((and (Xq Xu00)) ((or ((R Xu00) Xv0)) ((S Xu00) Xv0)))->(Xq Xv0)))->(Xq Xu))) (x5:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> (x4 x20))) as proof of (Xq Xu0)
% Found (((fun (P:Type) (x4:(((forall (Xu0:a) (Xv:a), (((and (Xq Xu0)) ((or ((R Xu0) Xv)) ((S Xu0) Xv)))->(Xq Xv)))->(Xq Xu))->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((R Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((S Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->P)))=> (((((and_rect ((forall (Xu0:a) (Xv0:a), (((and (Xq Xu0)) ((or ((R Xu0) Xv0)) ((S Xu0) Xv0)))->(Xq Xv0)))->(Xq Xu))) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((R Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((S Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))) P) x4) x3)) (Xq Xu0)) (fun (x4:((forall (Xu00:a) (Xv0:a), (((and (Xq Xu00)) ((or ((R Xu00) Xv0)) ((S Xu00) Xv0)))->(Xq Xv0)))->(Xq Xu))) (x5:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> (x4 x20))) as proof of (Xq Xu0)
% Found x40:=(x4 x00):(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found (x4 x00) as proof of (Xq Xu0)
% Found (x4 x00) as proof of (Xq Xu0)
% Found or_intror00:=(or_intror0 ((S Xx) Xw0)):(((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (or_intror0 ((S Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (forall (Xw0:a), (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0))))
% Found or_introl00:=(or_introl0 ((S Xx) Xw0)):(((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (or_introl0 ((S Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (forall (Xw0:a), (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0))))
% Found or_introl00:=(or_introl0 ((S Xx) Xw0)):(((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (or_introl0 ((S Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (forall (Xw0:a), (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0))))
% Found or_intror00:=(or_intror0 ((S Xx) Xw0)):(((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (or_intror0 ((S Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (forall (Xw0:a), (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0))))
% Found or_intror00:=(or_intror0 ((S Xx) Xw0)):(((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (or_intror0 ((S Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (forall (Xw0:a), (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0))))
% Found or_introl00:=(or_introl0 ((S Xx) Xw0)):(((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (or_introl0 ((S Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (forall (Xw0:a), (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0))))
% Found x5:((or ((R Xx) Xu)) ((S Xx) Xu))
% Found (fun (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5) as proof of ((or ((R Xx) Xu)) ((S Xx) Xu))
% Found (fun (x5:((or ((R Xx) Xu)) ((S Xx) Xu))) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5) as proof of (((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->((or ((R Xx) Xu)) ((S Xx) Xu)))
% Found (fun (x5:((or ((R Xx) Xu)) ((S Xx) Xu))) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5) as proof of (((or ((R Xx) Xu)) ((S Xx) Xu))->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->((or ((R Xx) Xu)) ((S Xx) Xu))))
% Found (and_rect10 (fun (x5:((or ((R Xx) Xu)) ((S Xx) Xu))) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5)) as proof of ((or ((R Xx) Xu)) ((S Xx) Xu))
% Found ((and_rect1 ((or ((R Xx) Xu)) ((S Xx) Xu))) (fun (x5:((or ((R Xx) Xu)) ((S Xx) Xu))) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5)) as proof of ((or ((R Xx) Xu)) ((S Xx) Xu))
% Found (((fun (P:Type) (x5:(((or ((R Xx) Xu)) ((S Xx) Xu))->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->P)))=> (((((and_rect ((or ((R Xx) Xu)) ((S Xx) Xu))) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))) P) x5) x4)) ((or ((R Xx) Xu)) ((S Xx) Xu))) (fun (x5:((or ((R Xx) Xu)) ((S Xx) Xu))) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5)) as proof of ((or ((R Xx) Xu)) ((S Xx) Xu))
% Found (fun (x4:((and ((or ((R Xx) Xu)) ((S Xx) Xu))) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))))=> (((fun (P:Type) (x5:(((or ((R Xx) Xu)) ((S Xx) Xu))->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->P)))=> (((((and_rect ((or ((R Xx) Xu)) ((S Xx) Xu))) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))) P) x5) x4)) ((or ((R Xx) Xu)) ((S Xx) Xu))) (fun (x5:((or ((R Xx) Xu)) ((S Xx) Xu))) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5))) as proof of ((or ((R Xx) Xu)) ((S Xx) Xu))
% Found (fun (Xv:a) (x4:((and ((or ((R Xx) Xu)) ((S Xx) Xu))) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))))=> (((fun (P:Type) (x5:(((or ((R Xx) Xu)) ((S Xx) Xu))->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->P)))=> (((((and_rect ((or ((R Xx) Xu)) ((S Xx) Xu))) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))) P) x5) x4)) ((or ((R Xx) Xu)) ((S Xx) Xu))) (fun (x5:((or ((R Xx) Xu)) ((S Xx) Xu))) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5))) as proof of (((and ((or ((R Xx) Xu)) ((S Xx) Xu))) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))->((or ((R Xx) Xu)) ((S Xx) Xu)))
% Found (fun (Xu0:a) (Xv:a) (x4:((and ((or ((R Xx) Xu)) ((S Xx) Xu))) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))))=> (((fun (P:Type) (x5:(((or ((R Xx) Xu)) ((S Xx) Xu))->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->P)))=> (((((and_rect ((or ((R Xx) Xu)) ((S Xx) Xu))) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))) P) x5) x4)) ((or ((R Xx) Xu)) ((S Xx) Xu))) (fun (x5:((or ((R Xx) Xu)) ((S Xx) Xu))) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5))) as proof of (forall (Xv:a), (((and ((or ((R Xx) Xu)) ((S Xx) Xu))) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))->((or ((R Xx) Xu)) ((S Xx) Xu))))
% Found (fun (Xu0:a) (Xv:a) (x4:((and ((or ((R Xx) Xu)) ((S Xx) Xu))) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))))=> (((fun (P:Type) (x5:(((or ((R Xx) Xu)) ((S Xx) Xu))->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->P)))=> (((((and_rect ((or ((R Xx) Xu)) ((S Xx) Xu))) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))) P) x5) x4)) ((or ((R Xx) Xu)) ((S Xx) Xu))) (fun (x5:((or ((R Xx) Xu)) ((S Xx) Xu))) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5))) as proof of (forall (Xu0:a) (Xv:a), (((and ((or ((R Xx) Xu)) ((S Xx) Xu))) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu0) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))->((or ((R Xx) Xu)) ((S Xx) Xu))))
% Found x4:(Xq0 Xv)
% Found (fun (x5:((or ((R Xu0) Xv0)) ((S Xu0) Xv0)))=> x4) as proof of (Xq0 Xv)
% Found (fun (x4:(Xq0 Xv)) (x5:((or ((R Xu0) Xv0)) ((S Xu0) Xv0)))=> x4) as proof of (((or ((R Xu0) Xv0)) ((S Xu0) Xv0))->(Xq0 Xv))
% Found (fun (x4:(Xq0 Xv)) (x5:((or ((R Xu0) Xv0)) ((S Xu0) Xv0)))=> x4) as proof of ((Xq0 Xv)->(((or ((R Xu0) Xv0)) ((S Xu0) Xv0))->(Xq0 Xv)))
% Found (and_rect10 (fun (x4:(Xq0 Xv)) (x5:((or ((R Xu0) Xv0)) ((S Xu0) Xv0)))=> x4)) as proof of (Xq0 Xv)
% Found ((and_rect1 (Xq0 Xv)) (fun (x4:(Xq0 Xv)) (x5:((or ((R Xu0) Xv0)) ((S Xu0) Xv0)))=> x4)) as proof of (Xq0 Xv)
% Found (((fun (P:Type) (x4:((Xq0 Xv)->(((or ((R Xu0) Xv0)) ((S Xu0) Xv0))->P)))=> (((((and_rect (Xq0 Xv)) ((or ((R Xu0) Xv0)) ((S Xu0) Xv0))) P) x4) x21)) (Xq0 Xv)) (fun (x4:(Xq0 Xv)) (x5:((or ((R Xu0) Xv0)) ((S Xu0) Xv0)))=> x4)) as proof of (Xq0 Xv)
% Found (fun (x21:((and (Xq0 Xv)) ((or ((R Xu0) Xv0)) ((S Xu0) Xv0))))=> (((fun (P:Type) (x4:((Xq0 Xv)->(((or ((R Xu0) Xv0)) ((S Xu0) Xv0))->P)))=> (((((and_rect (Xq0 Xv)) ((or ((R Xu0) Xv0)) ((S Xu0) Xv0))) P) x4) x21)) (Xq0 Xv)) (fun (x4:(Xq0 Xv)) (x5:((or ((R Xu0) Xv0)) ((S Xu0) Xv0)))=> x4))) as proof of (Xq0 Xv)
% Found (fun (Xv0:a) (x21:((and (Xq0 Xv)) ((or ((R Xu0) Xv0)) ((S Xu0) Xv0))))=> (((fun (P:Type) (x4:((Xq0 Xv)->(((or ((R Xu0) Xv0)) ((S Xu0) Xv0))->P)))=> (((((and_rect (Xq0 Xv)) ((or ((R Xu0) Xv0)) ((S Xu0) Xv0))) P) x4) x21)) (Xq0 Xv)) (fun (x4:(Xq0 Xv)) (x5:((or ((R Xu0) Xv0)) ((S Xu0) Xv0)))=> x4))) as proof of (((and (Xq0 Xv)) ((or ((R Xu0) Xv0)) ((S Xu0) Xv0)))->(Xq0 Xv))
% Found (fun (Xu0:a) (Xv0:a) (x21:((and (Xq0 Xv)) ((or ((R Xu0) Xv0)) ((S Xu0) Xv0))))=> (((fun (P:Type) (x4:((Xq0 Xv)->(((or ((R Xu0) Xv0)) ((S Xu0) Xv0))->P)))=> (((((and_rect (Xq0 Xv)) ((or ((R Xu0) Xv0)) ((S Xu0) Xv0))) P) x4) x21)) (Xq0 Xv)) (fun (x4:(Xq0 Xv)) (x5:((or ((R Xu0) Xv0)) ((S Xu0) Xv0)))=> x4))) as proof of (forall (Xv0:a), (((and (Xq0 Xv)) ((or ((R Xu0) Xv0)) ((S Xu0) Xv0)))->(Xq0 Xv)))
% Found (fun (Xu0:a) (Xv0:a) (x21:((and (Xq0 Xv)) ((or ((R Xu0) Xv0)) ((S Xu0) Xv0))))=> (((fun (P:Type) (x4:((Xq0 Xv)->(((or ((R Xu0) Xv0)) ((S Xu0) Xv0))->P)))=> (((((and_rect (Xq0 Xv)) ((or ((R Xu0) Xv0)) ((S Xu0) Xv0))) P) x4) x21)) (Xq0 Xv)) (fun (x4:(Xq0 Xv)) (x5:((or ((R Xu0) Xv0)) ((S Xu0) Xv0)))=> x4))) as proof of (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xv)) ((or ((R Xu0) Xv0)) ((S Xu0) Xv0)))->(Xq0 Xv)))
% Found or_intror00:=(or_intror0 ((S Xx) Xw0)):(((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (or_intror0 ((S Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (forall (Xw0:a), (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0))))
% Found or_introl00:=(or_introl0 ((S Xx) Xw0)):(((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (or_introl0 ((S Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (forall (Xw0:a), (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0))))
% Found or_introl00:=(or_introl0 ((R Xx) Xw0)):(((S Xx) Xw0)->((or ((S Xx) Xw0)) ((R Xx) Xw0)))
% Found (or_introl0 ((R Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((S Xx) Xw0)) ((R Xx) Xw0)))
% Found ((or_introl ((S Xx) Xw0)) ((R Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((S Xx) Xw0)) ((R Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((S Xx) Xw0)) ((R Xx) Xw0))) as proof of (((S Xx) Xw0)->((or ((S Xx) Xw0)) ((R Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((S Xx) Xw0)) ((R Xx) Xw0))) as proof of (forall (Xw0:a), (((S Xx) Xw0)->((or ((S Xx) Xw0)) ((R Xx) Xw0))))
% Found or_intror00:=(or_intror0 ((R Xx) Xw0)):(((R Xx) Xw0)->((or ((S Xx) Xw0)) ((R Xx) Xw0)))
% Found (or_intror0 ((R Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((S Xx) Xw0)) ((R Xx) Xw0)))
% Found ((or_intror ((S Xx) Xw0)) ((R Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((S Xx) Xw0)) ((R Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((S Xx) Xw0)) ((R Xx) Xw0))) as proof of (((R Xx) Xw0)->((or ((S Xx) Xw0)) ((R Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((S Xx) Xw0)) ((R Xx) Xw0))) as proof of (forall (Xw0:a), (((R Xx) Xw0)->((or ((S Xx) Xw0)) ((R Xx) Xw0))))
% Found x6:((R Xx) Xw0)
% Found (fun (x6:((R Xx) Xw0))=> x6) as proof of ((R Xx) Xw0)
% Found (fun (Xw0:a) (x6:((R Xx) Xw0))=> x6) as proof of (((R Xx) Xw0)->((R Xx) Xw0))
% Found (fun (Xw0:a) (x6:((R Xx) Xw0))=> x6) as proof of (forall (Xw0:a), (((R Xx) Xw0)->((R Xx) Xw0)))
% Found x6:((S Xx) Xw0)
% Found (fun (x6:((S Xx) Xw0))=> x6) as proof of ((S Xx) Xw0)
% Found (fun (Xw0:a) (x6:((S Xx) Xw0))=> x6) as proof of (((S Xx) Xw0)->((S Xx) Xw0))
% Found (fun (Xw0:a) (x6:((S Xx) Xw0))=> x6) as proof of (forall (Xw0:a), (((S Xx) Xw0)->((S Xx) Xw0)))
% Found x7:(Xq Xu)
% Found (fun (x8:((or ((R Xu) Xu0)) ((S Xu) Xu0)))=> x7) as proof of (Xq Xu)
% Found (fun (x7:(Xq Xu)) (x8:((or ((R Xu) Xu0)) ((S Xu) Xu0)))=> x7) as proof of (((or ((R Xu) Xu0)) ((S Xu) Xu0))->(Xq Xu))
% Found (fun (x7:(Xq Xu)) (x8:((or ((R Xu) Xu0)) ((S Xu) Xu0)))=> x7) as proof of ((Xq Xu)->(((or ((R Xu) Xu0)) ((S Xu) Xu0))->(Xq Xu)))
% Found (and_rect20 (fun (x7:(Xq Xu)) (x8:((or ((R Xu) Xu0)) ((S Xu) Xu0)))=> x7)) as proof of (Xq Xu)
% Found ((and_rect2 (Xq Xu)) (fun (x7:(Xq Xu)) (x8:((or ((R Xu) Xu0)) ((S Xu) Xu0)))=> x7)) as proof of (Xq Xu)
% Found (((fun (P:Type) (x7:((Xq Xu)->(((or ((R Xu) Xu0)) ((S Xu) Xu0))->P)))=> (((((and_rect (Xq Xu)) ((or ((R Xu) Xu0)) ((S Xu) Xu0))) P) x7) x5)) (Xq Xu)) (fun (x7:(Xq Xu)) (x8:((or ((R Xu) Xu0)) ((S Xu) Xu0)))=> x7)) as proof of (Xq Xu)
% Found (((fun (P:Type) (x7:((Xq Xu)->(((or ((R Xu) Xu0)) ((S Xu) Xu0))->P)))=> (((((and_rect (Xq Xu)) ((or ((R Xu) Xu0)) ((S Xu) Xu0))) P) x7) x5)) (Xq Xu)) (fun (x7:(Xq Xu)) (x8:((or ((R Xu) Xu0)) ((S Xu) Xu0)))=> x7)) as proof of (Xq Xu)
% Found x300:=(x30 x20):(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found (x30 x20) as proof of (Xq Xu0)
% Found ((x3 x10) x20) as proof of (Xq Xu0)
% Found (fun (x4:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> ((x3 x10) x20)) as proof of (Xq Xu0)
% Found (fun (x3:((forall (Xw:a), (((or ((R Xx) Xw)) ((S Xx) Xw))->(Xq Xw)))->((forall (Xu00:a) (Xv:a), (((and (Xq Xu00)) ((or ((R Xu00) Xv)) ((S Xu00) Xv)))->(Xq Xv)))->(Xq Xu)))) (x4:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> ((x3 x10) x20)) as proof of (((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->(Xq Xu0))
% Found (fun (x3:((forall (Xw:a), (((or ((R Xx) Xw)) ((S Xx) Xw))->(Xq Xw)))->((forall (Xu00:a) (Xv:a), (((and (Xq Xu00)) ((or ((R Xu00) Xv)) ((S Xu00) Xv)))->(Xq Xv)))->(Xq Xu)))) (x4:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> ((x3 x10) x20)) as proof of (((forall (Xw:a), (((or ((R Xx) Xw)) ((S Xx) Xw))->(Xq Xw)))->((forall (Xu00:a) (Xv:a), (((and (Xq Xu00)) ((or ((R Xu00) Xv)) ((S Xu00) Xv)))->(Xq Xv)))->(Xq Xu)))->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->(Xq Xu0)))
% Found (and_rect10 (fun (x3:((forall (Xw:a), (((or ((R Xx) Xw)) ((S Xx) Xw))->(Xq Xw)))->((forall (Xu00:a) (Xv:a), (((and (Xq Xu00)) ((or ((R Xu00) Xv)) ((S Xu00) Xv)))->(Xq Xv)))->(Xq Xu)))) (x4:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> ((x3 x10) x20))) as proof of (Xq Xu0)
% Found ((and_rect1 (Xq Xu0)) (fun (x3:((forall (Xw:a), (((or ((R Xx) Xw)) ((S Xx) Xw))->(Xq Xw)))->((forall (Xu00:a) (Xv:a), (((and (Xq Xu00)) ((or ((R Xu00) Xv)) ((S Xu00) Xv)))->(Xq Xv)))->(Xq Xu)))) (x4:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> ((x3 x10) x20))) as proof of (Xq Xu0)
% Found (((fun (P:Type) (x3:(((forall (Xw:a), (((or ((R Xx) Xw)) ((S Xx) Xw))->(Xq Xw)))->((forall (Xu0:a) (Xv:a), (((and (Xq Xu0)) ((or ((R Xu0) Xv)) ((S Xu0) Xv)))->(Xq Xv)))->(Xq Xu)))->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((R Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((S Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->P)))=> (((((and_rect ((forall (Xw:a), (((or ((R Xx) Xw)) ((S Xx) Xw))->(Xq Xw)))->((forall (Xu0:a) (Xv0:a), (((and (Xq Xu0)) ((or ((R Xu0) Xv0)) ((S Xu0) Xv0)))->(Xq Xv0)))->(Xq Xu)))) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((R Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((S Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))) P) x3) x2)) (Xq Xu0)) (fun (x3:((forall (Xw:a), (((or ((R Xx) Xw)) ((S Xx) Xw))->(Xq Xw)))->((forall (Xu00:a) (Xv0:a), (((and (Xq Xu00)) ((or ((R Xu00) Xv0)) ((S Xu00) Xv0)))->(Xq Xv0)))->(Xq Xu)))) (x4:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> ((x3 x10) x20))) as proof of (Xq Xu0)
% Found (((fun (P:Type) (x3:(((forall (Xw:a), (((or ((R Xx) Xw)) ((S Xx) Xw))->(Xq Xw)))->((forall (Xu0:a) (Xv:a), (((and (Xq Xu0)) ((or ((R Xu0) Xv)) ((S Xu0) Xv)))->(Xq Xv)))->(Xq Xu)))->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((R Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((S Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->P)))=> (((((and_rect ((forall (Xw:a), (((or ((R Xx) Xw)) ((S Xx) Xw))->(Xq Xw)))->((forall (Xu0:a) (Xv0:a), (((and (Xq Xu0)) ((or ((R Xu0) Xv0)) ((S Xu0) Xv0)))->(Xq Xv0)))->(Xq Xu)))) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((R Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((S Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))) P) x3) x2)) (Xq Xu0)) (fun (x3:((forall (Xw:a), (((or ((R Xx) Xw)) ((S Xx) Xw))->(Xq Xw)))->((forall (Xu00:a) (Xv0:a), (((and (Xq Xu00)) ((or ((R Xu00) Xv0)) ((S Xu00) Xv0)))->(Xq Xv0)))->(Xq Xu)))) (x4:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> ((x3 x10) x20))) as proof of (Xq Xu0)
% Found x5:(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found (fun (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5) as proof of (Xq Xu0)
% Found (fun (x5:(Xq Xu)) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5) as proof of (((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->(Xq Xu0))
% Found (fun (x5:(Xq Xu)) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5) as proof of ((Xq Xu)->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->(Xq Xu0)))
% Found (and_rect10 (fun (x5:(Xq Xu)) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5)) as proof of (Xq Xu0)
% Found ((and_rect1 (Xq Xu0)) (fun (x5:(Xq Xu)) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5)) as proof of (Xq Xu0)
% Found (((fun (P:Type) (x5:((Xq Xu)->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((R Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((S Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->P)))=> (((((and_rect (Xq Xu)) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((R Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((S Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))) P) x5) x2)) (Xq Xu0)) (fun (x5:(Xq Xu)) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5)) as proof of (Xq Xu0)
% Found (((fun (P:Type) (x5:((Xq Xu)->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((R Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((S Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->P)))=> (((((and_rect (Xq Xu)) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((R Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((S Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))) P) x5) x2)) (Xq Xu0)) (fun (x5:(Xq Xu)) (x6:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> x5)) as proof of (Xq Xu0)
% Found x5:(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found x5 as proof of (Xq Xu0)
% Found x5:(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found x5 as proof of (Xq Xu0)
% Found x5:(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found x5 as proof of (Xq Xu0)
% Found x5:(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found x5 as proof of (Xq Xu0)
% Found or_introl00:=(or_introl0 ((S Xx) Xw0)):(((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (or_introl0 ((S Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (forall (Xw0:a), (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0))))
% Found or_intror00:=(or_intror0 ((S Xx) Xw0)):(((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (or_intror0 ((S Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (forall (Xw0:a), (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0))))
% Found x5:(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found x5 as proof of (Xq Xu0)
% Found x5:(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found x5 as proof of (Xq Xu0)
% Found x5:(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found x5 as proof of (Xq Xu0)
% Found x5:(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found x5 as proof of (Xq Xu0)
% Found x40:=(x4 x00):(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found (x4 x00) as proof of (Xq Xu0)
% Found (fun (x5:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> (x4 x00)) as proof of (Xq Xu0)
% Found (fun (x4:(((and (forall (Xw:a), (((or ((R Xx) Xw)) ((S Xx) Xw))->(Xq Xw)))) (forall (Xu00:a) (Xv:a), (((and (Xq Xu00)) ((or ((R Xu00) Xv)) ((S Xu00) Xv)))->(Xq Xv))))->(Xq Xu))) (x5:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> (x4 x00)) as proof of (((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->(Xq Xu0))
% Found (fun (x4:(((and (forall (Xw:a), (((or ((R Xx) Xw)) ((S Xx) Xw))->(Xq Xw)))) (forall (Xu00:a) (Xv:a), (((and (Xq Xu00)) ((or ((R Xu00) Xv)) ((S Xu00) Xv)))->(Xq Xv))))->(Xq Xu))) (x5:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> (x4 x00)) as proof of ((((and (forall (Xw:a), (((or ((R Xx) Xw)) ((S Xx) Xw))->(Xq Xw)))) (forall (Xu00:a) (Xv:a), (((and (Xq Xu00)) ((or ((R Xu00) Xv)) ((S Xu00) Xv)))->(Xq Xv))))->(Xq Xu))->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->(Xq Xu0)))
% Found (and_rect10 (fun (x4:(((and (forall (Xw:a), (((or ((R Xx) Xw)) ((S Xx) Xw))->(Xq Xw)))) (forall (Xu00:a) (Xv:a), (((and (Xq Xu00)) ((or ((R Xu00) Xv)) ((S Xu00) Xv)))->(Xq Xv))))->(Xq Xu))) (x5:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> (x4 x00))) as proof of (Xq Xu0)
% Found ((and_rect1 (Xq Xu0)) (fun (x4:(((and (forall (Xw:a), (((or ((R Xx) Xw)) ((S Xx) Xw))->(Xq Xw)))) (forall (Xu00:a) (Xv:a), (((and (Xq Xu00)) ((or ((R Xu00) Xv)) ((S Xu00) Xv)))->(Xq Xv))))->(Xq Xu))) (x5:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> (x4 x00))) as proof of (Xq Xu0)
% Found (((fun (P:Type) (x4:((((and (forall (Xw:a), (((or ((R Xx) Xw)) ((S Xx) Xw))->(Xq Xw)))) (forall (Xu0:a) (Xv:a), (((and (Xq Xu0)) ((or ((R Xu0) Xv)) ((S Xu0) Xv)))->(Xq Xv))))->(Xq Xu))->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((R Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((S Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->P)))=> (((((and_rect (((and (forall (Xw:a), (((or ((R Xx) Xw)) ((S Xx) Xw))->(Xq Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq Xu0)) ((or ((R Xu0) Xv0)) ((S Xu0) Xv0)))->(Xq Xv0))))->(Xq Xu))) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((R Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((S Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))) P) x4) x1)) (Xq Xu0)) (fun (x4:(((and (forall (Xw:a), (((or ((R Xx) Xw)) ((S Xx) Xw))->(Xq Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq Xu00)) ((or ((R Xu00) Xv0)) ((S Xu00) Xv0)))->(Xq Xv0))))->(Xq Xu))) (x5:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> (x4 x00))) as proof of (Xq Xu0)
% Found (((fun (P:Type) (x4:((((and (forall (Xw:a), (((or ((R Xx) Xw)) ((S Xx) Xw))->(Xq Xw)))) (forall (Xu0:a) (Xv:a), (((and (Xq Xu0)) ((or ((R Xu0) Xv)) ((S Xu0) Xv)))->(Xq Xv))))->(Xq Xu))->(((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((R Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((S Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))->P)))=> (((((and_rect (((and (forall (Xw:a), (((or ((R Xx) Xw)) ((S Xx) Xw))->(Xq Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq Xu0)) ((or ((R Xu0) Xv0)) ((S Xu0) Xv0)))->(Xq Xv0))))->(Xq Xu))) ((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((R Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu0:a) (Xv0:a), (((and (Xq0 Xu0)) ((S Xu0) Xv0))->(Xq0 Xv0))))->(Xq0 Xv))))) P) x4) x1)) (Xq Xu0)) (fun (x4:(((and (forall (Xw:a), (((or ((R Xx) Xw)) ((S Xx) Xw))->(Xq Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq Xu00)) ((or ((R Xu00) Xv0)) ((S Xu00) Xv0)))->(Xq Xv0))))->(Xq Xu))) (x5:((or (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((R Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((R Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))) (forall (Xq0:(a->Prop)), (((and (forall (Xw:a), (((S Xu) Xw)->(Xq0 Xw)))) (forall (Xu00:a) (Xv0:a), (((and (Xq0 Xu00)) ((S Xu00) Xv0))->(Xq0 Xv0))))->(Xq0 Xv)))))=> (x4 x00))) as proof of (Xq Xu0)
% Found or_introl00:=(or_introl0 ((R Xx) Xw0)):(((S Xx) Xw0)->((or ((S Xx) Xw0)) ((R Xx) Xw0)))
% Found (or_introl0 ((R Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((S Xx) Xw0)) ((R Xx) Xw0)))
% Found ((or_introl ((S Xx) Xw0)) ((R Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((S Xx) Xw0)) ((R Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((S Xx) Xw0)) ((R Xx) Xw0))) as proof of (((S Xx) Xw0)->((or ((S Xx) Xw0)) ((R Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((S Xx) Xw0)) ((R Xx) Xw0))) as proof of (forall (Xw0:a), (((S Xx) Xw0)->((or ((S Xx) Xw0)) ((R Xx) Xw0))))
% Found or_intror00:=(or_intror0 ((R Xx) Xw0)):(((R Xx) Xw0)->((or ((S Xx) Xw0)) ((R Xx) Xw0)))
% Found (or_intror0 ((R Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((S Xx) Xw0)) ((R Xx) Xw0)))
% Found ((or_intror ((S Xx) Xw0)) ((R Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((S Xx) Xw0)) ((R Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((S Xx) Xw0)) ((R Xx) Xw0))) as proof of (((R Xx) Xw0)->((or ((S Xx) Xw0)) ((R Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((S Xx) Xw0)) ((R Xx) Xw0))) as proof of (forall (Xw0:a), (((R Xx) Xw0)->((or ((S Xx) Xw0)) ((R Xx) Xw0))))
% Found or_introl00:=(or_introl0 ((S Xx) Xw0)):(((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (or_introl0 ((S Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (forall (Xw0:a), (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0))))
% Found or_intror00:=(or_intror0 ((S Xx) Xw0)):(((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (or_intror0 ((S Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (forall (Xw0:a), (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0))))
% Found or_introl00:=(or_introl0 ((S Xx) Xw0)):(((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (or_introl0 ((S Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (forall (Xw0:a), (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0))))
% Found or_intror00:=(or_intror0 ((S Xx) Xw0)):(((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (or_intror0 ((S Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (forall (Xw0:a), (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0))))
% Found or_introl00:=(or_introl0 ((S Xx) Xw0)):(((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (or_introl0 ((S Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (forall (Xw0:a), (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0))))
% Found or_intror00:=(or_intror0 ((S Xx) Xw0)):(((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (or_intror0 ((S Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (forall (Xw0:a), (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0))))
% Found x5:(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found x5 as proof of (Xq Xu0)
% Found x5:(Xq Xu)
% Instantiate: Xu0:=Xu:a
% Found x5 as proof of (Xq Xu0)
% Found x6:((R Xx) Xw0)
% Found (fun (x6:((R Xx) Xw0))=> x6) as proof of ((R Xx) Xw0)
% Found (fun (Xw0:a) (x6:((R Xx) Xw0))=> x6) as proof of (((R Xx) Xw0)->((R Xx) Xw0))
% Found (fun (Xw0:a) (x6:((R Xx) Xw0))=> x6) as proof of (forall (Xw0:a), (((R Xx) Xw0)->((R Xx) Xw0)))
% Found x6:((S Xx) Xw0)
% Found (fun (x6:((S Xx) Xw0))=> x6) as proof of ((S Xx) Xw0)
% Found (fun (Xw0:a) (x6:((S Xx) Xw0))=> x6) as proof of (((S Xx) Xw0)->((S Xx) Xw0))
% Found (fun (Xw0:a) (x6:((S Xx) Xw0))=> x6) as proof of (forall (Xw0:a), (((S Xx) Xw0)->((S Xx) Xw0)))
% Found or_introl00:=(or_introl0 ((S Xx) Xw0)):(((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (or_introl0 ((S Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0)) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_introl ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (forall (Xw0:a), (((R Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0))))
% Found or_intror00:=(or_intror0 ((S Xx) Xw0)):(((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (or_intror0 ((S Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0)) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0)))
% Found (fun (Xw0:a)=> ((or_intror ((R Xx) Xw0)) ((S Xx) Xw0))) as proof of (forall (Xw0:a), (((S Xx) Xw0)->((or ((R Xx) Xw0)) ((S Xx) Xw0))))
% Found x5:(Xq X
% EOF
%------------------------------------------------------------------------------