%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV126^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n094.star.cs.uiowa.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory   : 32286.75MB
% OS       : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:46 EDT 2014

% Result   : Theorem 74.27s
% Output   : Proof 74.27s
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----ERROR: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV126^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n094.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 08:10:26 CDT 2014
% % CPUTime  : 74.27 
% 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 0x1618830>, <kernel.Type object at 0x1618f38>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (P:((a->(a->Prop))->Prop)) (R:(a->(a->Prop))) (S:(a->(a->Prop))) (Xx:a) (Xy:a), ((forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy)))->(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy))))) of role conjecture named cTHM252B_pme
% Conjecture to prove = (forall (P:((a->(a->Prop))->Prop)) (R:(a->(a->Prop))) (S:(a->(a->Prop))) (Xx:a) (Xy:a), ((forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy)))->(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (P:((a->(a->Prop))->Prop)) (R:(a->(a->Prop))) (S:(a->(a->Prop))) (Xx:a) (Xy:a), ((forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy)))->(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy)))))']
% Parameter a:Type.
% Trying to prove (forall (P:((a->(a->Prop))->Prop)) (R:(a->(a->Prop))) (S:(a->(a->Prop))) (Xx:a) (Xy:a), ((forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy)))->(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy)))))
% Found x2:(P Xp)
% Found x2 as proof of (P Xp)
% Found x3:(P Xp)
% Found x3 as proof of (P Xp)
% Found x3:(P Xp)
% Found (fun (x3:(P Xp))=> x3) as proof of (P Xp)
% Found (fun (x2:(forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (x3:(P Xp))=> x3) as proof of ((P Xp)->(P Xp))
% Found (fun (x2:(forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (x3:(P Xp))=> x3) as proof of ((forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))->((P Xp)->(P Xp)))
% Found (and_rect00 (fun (x2:(forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (x3:(P Xp))=> x3)) as proof of (P Xp)
% Found ((and_rect0 (P Xp)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (x3:(P Xp))=> x3)) as proof of (P Xp)
% Found (((fun (P0:Type) (x2:((forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))->((P Xp)->P0)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp)) P0) x2) x0)) (P Xp)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (x3:(P Xp))=> x3)) as proof of (P Xp)
% Found (((fun (P0:Type) (x2:((forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))->((P Xp)->P0)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp)) P0) x2) x0)) (P Xp)) (fun (x2:(forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (x3:(P Xp))=> x3)) as proof of (P Xp)
% Found x5:(P Xp0)
% Found x5 as proof of (P Xp0)
% Found x5:(P Xp0)
% Found x5 as proof of (P Xp0)
% Found x6:(P Xp0)
% Found x6 as proof of (P Xp0)
% Found x6:(P Xp0)
% Found x6 as proof of (P Xp0)
% Found x6:(P Xp0)
% Found (fun (x6:(P Xp0))=> x6) as proof of (P Xp0)
% Found (fun (x5:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x6:(P Xp0))=> x6) as proof of ((P Xp0)->(P Xp0))
% Found (fun (x5:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x6:(P Xp0))=> x6) as proof of ((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->(P Xp0)))
% Found (and_rect10 (fun (x5:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x6:(P Xp0))=> x6)) as proof of (P Xp0)
% Found ((and_rect1 (P Xp0)) (fun (x5:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x6:(P Xp0))=> x6)) as proof of (P Xp0)
% Found (((fun (P0:Type) (x5:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x5) x3)) (P Xp0)) (fun (x5:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x6:(P Xp0))=> x6)) as proof of (P Xp0)
% Found (((fun (P0:Type) (x5:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x5) x3)) (P Xp0)) (fun (x5:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x6:(P Xp0))=> x6)) as proof of (P Xp0)
% Found x6:(P Xp0)
% Found (fun (x6:(P Xp0))=> x6) as proof of (P Xp0)
% Found (fun (x5:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x6:(P Xp0))=> x6) as proof of ((P Xp0)->(P Xp0))
% Found (fun (x5:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x6:(P Xp0))=> x6) as proof of ((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->(P Xp0)))
% Found (and_rect10 (fun (x5:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x6:(P Xp0))=> x6)) as proof of (P Xp0)
% Found ((and_rect1 (P Xp0)) (fun (x5:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x6:(P Xp0))=> x6)) as proof of (P Xp0)
% Found (((fun (P0:Type) (x5:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x5) x3)) (P Xp0)) (fun (x5:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x6:(P Xp0))=> x6)) as proof of (P Xp0)
% Found (((fun (P0:Type) (x5:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x5) x3)) (P Xp0)) (fun (x5:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x6:(P Xp0))=> x6)) as proof of (P Xp0)
% Found x5:(P Xp0)
% Found x5 as proof of (P Xp0)
% Found x5:(P Xp0)
% Found x5 as proof of (P Xp0)
% Found x6:(P Xp0)
% Found x6 as proof of (P Xp0)
% Found x6:(P Xp0)
% Found x6 as proof of (P Xp0)
% Found x500:=(x50 Xy0):(((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x50 Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x500:=(x50 Xy0):(((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x50 Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x400:=(x40 Xy0):(((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x40 Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x4 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x4 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x400:=(x40 Xy0):(((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x40 Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x4 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x4 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x500:=(x50 Xy0):(((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x50 Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x500:=(x50 Xy0):(((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x50 Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x400:=(x40 Xy0):(((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x40 Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x4 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x4 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x400:=(x40 Xy0):(((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x40 Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x4 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x4 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x500:=(x50 Xy0):(((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x50 Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x500:=(x50 Xy0):(((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x50 Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x500:=(x50 Xy0):(((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x50 Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x500:=(x50 Xy0):(((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x50 Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x600:=(x60 Xy0):(((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x60 Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x600:=(x60 Xy0):(((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x60 Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x600:=(x60 Xy0):(((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x60 Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x600:=(x60 Xy0):(((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x60 Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x6:(P Xp0)
% Found (fun (x6:(P Xp0))=> x6) as proof of (P Xp0)
% Found (fun (x5:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x6:(P Xp0))=> x6) as proof of ((P Xp0)->(P Xp0))
% Found (fun (x5:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x6:(P Xp0))=> x6) as proof of ((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->(P Xp0)))
% Found (and_rect10 (fun (x5:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x6:(P Xp0))=> x6)) as proof of (P Xp0)
% Found ((and_rect1 (P Xp0)) (fun (x5:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x6:(P Xp0))=> x6)) as proof of (P Xp0)
% Found (((fun (P0:Type) (x5:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x5) x3)) (P Xp0)) (fun (x5:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x6:(P Xp0))=> x6)) as proof of (P Xp0)
% Found (((fun (P0:Type) (x5:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x5) x3)) (P Xp0)) (fun (x5:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x6:(P Xp0))=> x6)) as proof of (P Xp0)
% Found x6:(P Xp0)
% Found (fun (x6:(P Xp0))=> x6) as proof of (P Xp0)
% Found (fun (x5:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x6:(P Xp0))=> x6) as proof of ((P Xp0)->(P Xp0))
% Found (fun (x5:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x6:(P Xp0))=> x6) as proof of ((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->(P Xp0)))
% Found (and_rect10 (fun (x5:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x6:(P Xp0))=> x6)) as proof of (P Xp0)
% Found ((and_rect1 (P Xp0)) (fun (x5:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x6:(P Xp0))=> x6)) as proof of (P Xp0)
% Found (((fun (P0:Type) (x5:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x5) x3)) (P Xp0)) (fun (x5:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x6:(P Xp0))=> x6)) as proof of (P Xp0)
% Found (((fun (P0:Type) (x5:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x5) x3)) (P Xp0)) (fun (x5:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x6:(P Xp0))=> x6)) as proof of (P Xp0)
% Found x600:=(x60 Xy0):(((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x60 Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x600:=(x60 Xy0):(((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x60 Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x600:=(x60 Xy0):(((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x60 Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x600:=(x60 Xy0):(((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x60 Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x600:=(x60 Xy0):(((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x60 Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x600:=(x60 Xy0):(((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x60 Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x500:=(x50 Xy0):(((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x50 Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x500:=(x50 Xy0):(((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x50 Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x400:=(x40 Xy0):(((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x40 Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x4 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x4 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x400:=(x40 Xy0):(((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x40 Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x4 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x4 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x500:=(x50 Xy0):(((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x50 Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x500:=(x50 Xy0):(((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x50 Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x600:=(x60 Xy0):(((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x60 Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x600:=(x60 Xy0):(((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x60 Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x7:((R Xx0) Xy0)
% Found (fun (x7:((R Xx0) Xy0))=> x7) as proof of ((R Xx0) Xy0)
% Found (fun (x7:((R Xx0) Xy0))=> x7) as proof of (((R Xx0) Xy0)->((R Xx0) Xy0))
% Found x7:((S Xx0) Xy0)
% Found (fun (x7:((S Xx0) Xy0))=> x7) as proof of ((S Xx0) Xy0)
% Found (fun (x7:((S Xx0) Xy0))=> x7) as proof of (((S Xx0) Xy0)->((S Xx0) Xy0))
% Found x400:=(x40 Xy0):(((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x40 Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x4 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x4 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x400:=(x40 Xy0):(((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x40 Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x4 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x4 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x8:((S Xx0) Xy0)
% Found (fun (x8:((S Xx0) Xy0))=> x8) as proof of ((S Xx0) Xy0)
% Found (fun (x8:((S Xx0) Xy0))=> x8) as proof of (((S Xx0) Xy0)->((S Xx0) Xy0))
% Found x8:((R Xx0) Xy0)
% Found (fun (x8:((R Xx0) Xy0))=> x8) as proof of ((R Xx0) Xy0)
% Found (fun (x8:((R Xx0) Xy0))=> x8) as proof of (((R Xx0) Xy0)->((R Xx0) Xy0))
% Found x400:=(x40 Xy0):(((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x40 Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x4 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x4 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x400:=(x40 Xy0):(((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x40 Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x4 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x4 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x4000:=(x400 Xy0):(((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x400 Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x40 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x40 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x4000:=(x400 Xy0):(((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x400 Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x40 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x40 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x8:((S Xx0) Xy0)
% Found (fun (x8:((S Xx0) Xy0))=> x8) as proof of ((S Xx0) Xy0)
% Found (fun (x8:((S Xx0) Xy0))=> x8) as proof of (((S Xx0) Xy0)->((S Xx0) Xy0))
% Found x8:((R Xx0) Xy0)
% Found (fun (x8:((R Xx0) Xy0))=> x8) as proof of ((R Xx0) Xy0)
% Found (fun (x8:((R Xx0) Xy0))=> x8) as proof of (((R Xx0) Xy0)->((R Xx0) Xy0))
% Found x500:=(x50 Xy0):(((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x50 Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x500:=(x50 Xy0):(((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x50 Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x8:((R Xx0) Xy0)
% Found (fun (x8:((R Xx0) Xy0))=> x8) as proof of ((R Xx0) Xy0)
% Found (fun (x8:((R Xx0) Xy0))=> x8) as proof of (((R Xx0) Xy0)->((R Xx0) Xy0))
% Found x8:((S Xx0) Xy0)
% Found (fun (x8:((S Xx0) Xy0))=> x8) as proof of ((S Xx0) Xy0)
% Found (fun (x8:((S Xx0) Xy0))=> x8) as proof of (((S Xx0) Xy0)->((S Xx0) Xy0))
% Found x8:((S Xx0) Xy0)
% Found (fun (x8:((S Xx0) Xy0))=> x8) as proof of ((S Xx0) Xy0)
% Found (fun (x8:((S Xx0) Xy0))=> x8) as proof of (((S Xx0) Xy0)->((S Xx0) Xy0))
% Found x8:((R Xx0) Xy0)
% Found (fun (x8:((R Xx0) Xy0))=> x8) as proof of ((R Xx0) Xy0)
% Found (fun (x8:((R Xx0) Xy0))=> x8) as proof of (((R Xx0) Xy0)->((R Xx0) Xy0))
% Found x600:=(x60 Xy0):(((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x60 Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x600:=(x60 Xy0):(((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x60 Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x500:=(x50 Xy0):(((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x50 Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x500:=(x50 Xy0):(((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x50 Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x7:((R Xx0) Xy0)
% Found (fun (x7:((R Xx0) Xy0))=> x7) as proof of ((R Xx0) Xy0)
% Found (fun (x7:((R Xx0) Xy0))=> x7) as proof of (((R Xx0) Xy0)->((R Xx0) Xy0))
% Found x7:((S Xx0) Xy0)
% Found (fun (x7:((S Xx0) Xy0))=> x7) as proof of ((S Xx0) Xy0)
% Found (fun (x7:((S Xx0) Xy0))=> x7) as proof of (((S Xx0) Xy0)->((S Xx0) Xy0))
% Found x600:=(x60 Xy0):(((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x60 Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x600:=(x60 Xy0):(((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x60 Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x8:((R Xx0) Xy0)
% Found (fun (x8:((R Xx0) Xy0))=> x8) as proof of ((R Xx0) Xy0)
% Found (fun (x8:((R Xx0) Xy0))=> x8) as proof of (((R Xx0) Xy0)->((R Xx0) Xy0))
% Found x8:((S Xx0) Xy0)
% Found (fun (x8:((S Xx0) Xy0))=> x8) as proof of ((S Xx0) Xy0)
% Found (fun (x8:((S Xx0) Xy0))=> x8) as proof of (((S Xx0) Xy0)->((S Xx0) Xy0))
% Found x8:((S Xx0) Xy0)
% Found (fun (x8:((S Xx0) Xy0))=> x8) as proof of ((S Xx0) Xy0)
% Found (fun (x8:((S Xx0) Xy0))=> x8) as proof of (((S Xx0) Xy0)->((S Xx0) Xy0))
% Found x8:((R Xx0) Xy0)
% Found (fun (x8:((R Xx0) Xy0))=> x8) as proof of ((R Xx0) Xy0)
% Found (fun (x8:((R Xx0) Xy0))=> x8) as proof of (((R Xx0) Xy0)->((R Xx0) Xy0))
% Found x600:=(x60 Xy0):(((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x60 Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x600:=(x60 Xy0):(((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x60 Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x8:((R Xx0) Xy0)
% Found (fun (x8:((R Xx0) Xy0))=> x8) as proof of ((R Xx0) Xy0)
% Found (fun (x8:((R Xx0) Xy0))=> x8) as proof of (((R Xx0) Xy0)->((R Xx0) Xy0))
% Found x8:((S Xx0) Xy0)
% Found (fun (x8:((S Xx0) Xy0))=> x8) as proof of ((S Xx0) Xy0)
% Found (fun (x8:((S Xx0) Xy0))=> x8) as proof of (((S Xx0) Xy0)->((S Xx0) Xy0))
% Found x600:=(x60 Xy0):(((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x60 Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x600:=(x60 Xy0):(((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x60 Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x8:((S Xx0) Xy0)
% Found (fun (x8:((S Xx0) Xy0))=> x8) as proof of ((S Xx0) Xy0)
% Found (fun (x8:((S Xx0) Xy0))=> x8) as proof of (((S Xx0) Xy0)->((S Xx0) Xy0))
% Found x8:((R Xx0) Xy0)
% Found (fun (x8:((R Xx0) Xy0))=> x8) as proof of ((R Xx0) Xy0)
% Found (fun (x8:((R Xx0) Xy0))=> x8) as proof of (((R Xx0) Xy0)->((R Xx0) Xy0))
% Found x500:=(x50 Xy0):(((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x50 Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x500:=(x50 Xy0):(((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x50 Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x600:=(x60 Xy0):(((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x60 Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x600:=(x60 Xy0):(((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x60 Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x8:((S Xx0) Xy0)
% Found (fun (x8:((S Xx0) Xy0))=> x8) as proof of ((S Xx0) Xy0)
% Found (fun (x8:((S Xx0) Xy0))=> x8) as proof of (((S Xx0) Xy0)->((S Xx0) Xy0))
% Found x8:((R Xx0) Xy0)
% Found (fun (x8:((R Xx0) Xy0))=> x8) as proof of ((R Xx0) Xy0)
% Found (fun (x8:((R Xx0) Xy0))=> x8) as proof of (((R Xx0) Xy0)->((R Xx0) Xy0))
% Found x8:((R Xx0) Xy0)
% Found (fun (x8:((R Xx0) Xy0))=> x8) as proof of ((R Xx0) Xy0)
% Found (fun (x8:((R Xx0) Xy0))=> x8) as proof of (((R Xx0) Xy0)->((R Xx0) Xy0))
% Found x8:((S Xx0) Xy0)
% Found (fun (x8:((S Xx0) Xy0))=> x8) as proof of ((S Xx0) Xy0)
% Found (fun (x8:((S Xx0) Xy0))=> x8) as proof of (((S Xx0) Xy0)->((S Xx0) Xy0))
% Found x500:=(x50 Xy0):(((S Xx0) Xy0)->((Xp00 Xx0) Xy0))
% Found (x50 Xy0) as proof of (((S Xx0) Xy0)->((Xp00 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp00 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp00 Xx0) Xy0))
% Found x500:=(x50 Xy0):(((R Xx0) Xy0)->((Xp00 Xx0) Xy0))
% Found (x50 Xy0) as proof of (((R Xx0) Xy0)->((Xp00 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp00 Xx0) Xy0))
% Found ((x5 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp00 Xx0) Xy0))
% Found x7:((R Xx0) Xy0)
% Found (fun (x7:((R Xx0) Xy0))=> x7) as proof of ((R Xx0) Xy0)
% Found (fun (x7:((R Xx0) Xy0))=> x7) as proof of (((R Xx0) Xy0)->((R Xx0) Xy0))
% Found x7:((S Xx0) Xy0)
% Found (fun (x7:((S Xx0) Xy0))=> x7) as proof of ((S Xx0) Xy0)
% Found (fun (x7:((S Xx0) Xy0))=> x7) as proof of (((S Xx0) Xy0)->((S Xx0) Xy0))
% Found x400:=(x40 Xy0):(((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x40 Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x4 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x4 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x400:=(x40 Xy0):(((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x40 Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x4 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x4 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x600:=(x60 Xy0):(((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x60 Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x600:=(x60 Xy0):(((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x60 Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x6 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x8:((S Xx0) Xy0)
% Found (fun (x8:((S Xx0) Xy0))=> x8) as proof of ((S Xx0) Xy0)
% Found (fun (x8:((S Xx0) Xy0))=> x8) as proof of (((S Xx0) Xy0)->((S Xx0) Xy0))
% Found x8:((R Xx0) Xy0)
% Found (fun (x8:((R Xx0) Xy0))=> x8) as proof of ((R Xx0) Xy0)
% Found (fun (x8:((R Xx0) Xy0))=> x8) as proof of (((R Xx0) Xy0)->((R Xx0) Xy0))
% Found x4000:=(x400 Xy0):(((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x400 Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x40 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x40 Xx0) Xy0) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x4000:=(x400 Xy0):(((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found (x400 Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x40 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found ((x40 Xx0) Xy0) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x6000:=(x600 x5):((Xp0 Xx0) Xy0)
% Found (x600 x5) as proof of ((Xp0 Xx0) Xy0)
% Found ((x60 Xy0) x5) as proof of ((Xp0 Xx0) Xy0)
% Found (((x6 Xx0) Xy0) x5) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x7:(P Xp0))=> (((x6 Xx0) Xy0) x5)) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x5)) as proof of ((P Xp0)->((Xp0 Xx0) Xy0))
% Found (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x5)) as proof of ((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->((Xp0 Xx0) Xy0)))
% Found (and_rect10 (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x5))) as proof of ((Xp0 Xx0) Xy0)
% Found ((and_rect1 ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x5))) as proof of ((Xp0 Xx0) Xy0)
% Found (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x4)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x5))) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x5:((S Xx0) Xy0))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x4)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x5)))) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x5:((S Xx0) Xy0))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x4)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x5)))) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x6000:=(x600 x5):((Xp0 Xx0) Xy0)
% Found (x600 x5) as proof of ((Xp0 Xx0) Xy0)
% Found ((x60 Xy0) x5) as proof of ((Xp0 Xx0) Xy0)
% Found (((x6 Xx0) Xy0) x5) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x7:(P Xp0))=> (((x6 Xx0) Xy0) x5)) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x5)) as proof of ((P Xp0)->((Xp0 Xx0) Xy0))
% Found (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x5)) as proof of ((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->((Xp0 Xx0) Xy0)))
% Found (and_rect10 (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x5))) as proof of ((Xp0 Xx0) Xy0)
% Found ((and_rect1 ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x5))) as proof of ((Xp0 Xx0) Xy0)
% Found (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x4)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x5))) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x5:((R Xx0) Xy0))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x4)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x5)))) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x5:((R Xx0) Xy0))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x4)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x5)))) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x8:((S Xx0) Xy0)
% Found (fun (x8:((S Xx0) Xy0))=> x8) as proof of ((S Xx0) Xy0)
% Found (fun (x8:((S Xx0) Xy0))=> x8) as proof of (((S Xx0) Xy0)->((S Xx0) Xy0))
% Found x8:((R Xx0) Xy0)
% Found (fun (x8:((R Xx0) Xy0))=> x8) as proof of ((R Xx0) Xy0)
% Found (fun (x8:((R Xx0) Xy0))=> x8) as proof of (((R Xx0) Xy0)->((R Xx0) Xy0))
% Found x8:((R Xx0) Xy0)
% Found (fun (x8:((R Xx0) Xy0))=> x8) as proof of ((R Xx0) Xy0)
% Found (fun (x8:((R Xx0) Xy0))=> x8) as proof of (((R Xx0) Xy0)->((R Xx0) Xy0))
% Found x8:((S Xx0) Xy0)
% Found (fun (x8:((S Xx0) Xy0))=> x8) as proof of ((S Xx0) Xy0)
% Found (fun (x8:((S Xx0) Xy0))=> x8) as proof of (((S Xx0) Xy0)->((S Xx0) Xy0))
% Found x7000:=(x700 x6):((Xp0 Xx0) Xy0)
% Found (x700 x6) as proof of ((Xp0 Xx0) Xy0)
% Found ((x70 Xy0) x6) as proof of ((Xp0 Xx0) Xy0)
% Found (((x7 Xx0) Xy0) x6) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x7:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)) as proof of ((P Xp0)->((Xp0 Xx0) Xy0))
% Found (fun (x7:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)) as proof of ((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->((Xp0 Xx0) Xy0)))
% Found (and_rect10 (fun (x7:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6))) as proof of ((Xp0 Xx0) Xy0)
% Found ((and_rect1 ((Xp0 Xx0) Xy0)) (fun (x7:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6))) as proof of ((Xp0 Xx0) Xy0)
% Found (((fun (P0:Type) (x7:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x7) x3)) ((Xp0 Xx0) Xy0)) (fun (x7:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6))) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x6:((S Xx0) Xy0))=> (((fun (P0:Type) (x7:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x7) x3)) ((Xp0 Xx0) Xy0)) (fun (x7:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)))) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x6:((S Xx0) Xy0))=> (((fun (P0:Type) (x7:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x7) x3)) ((Xp0 Xx0) Xy0)) (fun (x7:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)))) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x7000:=(x700 x6):((Xp0 Xx0) Xy0)
% Found (x700 x6) as proof of ((Xp0 Xx0) Xy0)
% Found ((x70 Xy0) x6) as proof of ((Xp0 Xx0) Xy0)
% Found (((x7 Xx0) Xy0) x6) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x7:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)) as proof of ((P Xp0)->((Xp0 Xx0) Xy0))
% Found (fun (x7:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)) as proof of ((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->((Xp0 Xx0) Xy0)))
% Found (and_rect10 (fun (x7:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6))) as proof of ((Xp0 Xx0) Xy0)
% Found ((and_rect1 ((Xp0 Xx0) Xy0)) (fun (x7:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6))) as proof of ((Xp0 Xx0) Xy0)
% Found (((fun (P0:Type) (x7:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x7) x3)) ((Xp0 Xx0) Xy0)) (fun (x7:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6))) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x6:((R Xx0) Xy0))=> (((fun (P0:Type) (x7:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x7) x3)) ((Xp0 Xx0) Xy0)) (fun (x7:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)))) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x6:((R Xx0) Xy0))=> (((fun (P0:Type) (x7:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x7) x3)) ((Xp0 Xx0) Xy0)) (fun (x7:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)))) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x7000:=(x700 x6):((Xp0 Xx0) Xy0)
% Found (x700 x6) as proof of ((Xp0 Xx0) Xy0)
% Found ((x70 Xy0) x6) as proof of ((Xp0 Xx0) Xy0)
% Found (((x7 Xx0) Xy0) x6) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x7:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)) as proof of ((P Xp0)->((Xp0 Xx0) Xy0))
% Found (fun (x7:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)) as proof of ((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->((Xp0 Xx0) Xy0)))
% Found (and_rect10 (fun (x7:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6))) as proof of ((Xp0 Xx0) Xy0)
% Found ((and_rect1 ((Xp0 Xx0) Xy0)) (fun (x7:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6))) as proof of ((Xp0 Xx0) Xy0)
% Found (((fun (P0:Type) (x7:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x7) x5)) ((Xp0 Xx0) Xy0)) (fun (x7:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6))) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x6:((R Xx0) Xy0))=> (((fun (P0:Type) (x7:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x7) x5)) ((Xp0 Xx0) Xy0)) (fun (x7:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)))) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x6:((R Xx0) Xy0))=> (((fun (P0:Type) (x7:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x7) x5)) ((Xp0 Xx0) Xy0)) (fun (x7:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)))) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x7000:=(x700 x6):((Xp0 Xx0) Xy0)
% Found (x700 x6) as proof of ((Xp0 Xx0) Xy0)
% Found ((x70 Xy0) x6) as proof of ((Xp0 Xx0) Xy0)
% Found (((x7 Xx0) Xy0) x6) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x7:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)) as proof of ((P Xp0)->((Xp0 Xx0) Xy0))
% Found (fun (x7:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)) as proof of ((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->((Xp0 Xx0) Xy0)))
% Found (and_rect10 (fun (x7:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6))) as proof of ((Xp0 Xx0) Xy0)
% Found ((and_rect1 ((Xp0 Xx0) Xy0)) (fun (x7:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6))) as proof of ((Xp0 Xx0) Xy0)
% Found (((fun (P0:Type) (x7:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x7) x5)) ((Xp0 Xx0) Xy0)) (fun (x7:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6))) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x6:((S Xx0) Xy0))=> (((fun (P0:Type) (x7:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x7) x5)) ((Xp0 Xx0) Xy0)) (fun (x7:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)))) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x6:((S Xx0) Xy0))=> (((fun (P0:Type) (x7:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x7) x5)) ((Xp0 Xx0) Xy0)) (fun (x7:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)))) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x8:((S Xx0) Xy0)
% Found (fun (x8:((S Xx0) Xy0))=> x8) as proof of ((S Xx0) Xy0)
% Found (fun (x8:((S Xx0) Xy0))=> x8) as proof of (((S Xx0) Xy0)->((S Xx0) Xy0))
% Found x8:((R Xx0) Xy0)
% Found (fun (x8:((R Xx0) Xy0))=> x8) as proof of ((R Xx0) Xy0)
% Found (fun (x8:((R Xx0) Xy0))=> x8) as proof of (((R Xx0) Xy0)->((R Xx0) Xy0))
% Found x7000:=(x700 x6):((Xp0 Xx0) Xy0)
% Found (x700 x6) as proof of ((Xp0 Xx0) Xy0)
% Found ((x70 Xy0) x6) as proof of ((Xp0 Xx0) Xy0)
% Found (((x7 Xx0) Xy0) x6) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x7:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)) as proof of ((P Xp0)->((Xp0 Xx0) Xy0))
% Found (fun (x7:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)) as proof of ((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->((Xp0 Xx0) Xy0)))
% Found (and_rect10 (fun (x7:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6))) as proof of ((Xp0 Xx0) Xy0)
% Found ((and_rect1 ((Xp0 Xx0) Xy0)) (fun (x7:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6))) as proof of ((Xp0 Xx0) Xy0)
% Found (((fun (P0:Type) (x7:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x7) x5)) ((Xp0 Xx0) Xy0)) (fun (x7:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6))) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x6:((S Xx0) Xy0))=> (((fun (P0:Type) (x7:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x7) x5)) ((Xp0 Xx0) Xy0)) (fun (x7:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)))) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x6:((S Xx0) Xy0))=> (((fun (P0:Type) (x7:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x7) x5)) ((Xp0 Xx0) Xy0)) (fun (x7:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)))) as proof of (((S Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x7000:=(x700 x6):((Xp0 Xx0) Xy0)
% Found (x700 x6) as proof of ((Xp0 Xx0) Xy0)
% Found ((x70 Xy0) x6) as proof of ((Xp0 Xx0) Xy0)
% Found (((x7 Xx0) Xy0) x6) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x7:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)) as proof of ((P Xp0)->((Xp0 Xx0) Xy0))
% Found (fun (x7:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)) as proof of ((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->((Xp0 Xx0) Xy0)))
% Found (and_rect10 (fun (x7:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6))) as proof of ((Xp0 Xx0) Xy0)
% Found ((and_rect1 ((Xp0 Xx0) Xy0)) (fun (x7:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6))) as proof of ((Xp0 Xx0) Xy0)
% Found (((fun (P0:Type) (x7:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x7) x5)) ((Xp0 Xx0) Xy0)) (fun (x7:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6))) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x6:((R Xx0) Xy0))=> (((fun (P0:Type) (x7:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x7) x5)) ((Xp0 Xx0) Xy0)) (fun (x7:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)))) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x6:((R Xx0) Xy0))=> (((fun (P0:Type) (x7:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x7) x5)) ((Xp0 Xx0) Xy0)) (fun (x7:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x8:(P Xp0))=> (((x7 Xx0) Xy0) x6)))) as proof of (((R Xx0) Xy0)->((Xp0 Xx0) Xy0))
% Found x6000:=(x600 x4):((Xp0 Xx0) Xy0)
% Found (x600 x4) as proof of ((Xp0 Xx0) Xy0)
% Found ((x60 Xy0) x4) as proof of ((Xp0 Xx0) Xy0)
% Found (((x6 Xx0) Xy0) x4) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)) as proof of ((P Xp0)->((Xp0 Xx0) Xy0))
% Found (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)) as proof of ((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->((Xp0 Xx0) Xy0)))
% Found (and_rect10 (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))) as proof of ((Xp0 Xx0) Xy0)
% Found ((and_rect1 ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))) as proof of ((Xp0 Xx0) Xy0)
% Found (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)))) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)))) as proof of (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))
% Found (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)))) as proof of (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))
% Found (or_intror00 (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))) as proof of ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))
% Found ((or_intror0 (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))) as proof of ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))
% Found (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))) as proof of ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))
% Found (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)))))) as proof of ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))
% Found (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)))))) as proof of (((S Xx0) Xy0)->((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))))
% Found x6000:=(x600 x4):((Xp0 Xx0) Xy0)
% Found (x600 x4) as proof of ((Xp0 Xx0) Xy0)
% Found ((x60 Xy0) x4) as proof of ((Xp0 Xx0) Xy0)
% Found (((x6 Xx0) Xy0) x4) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)) as proof of ((P Xp0)->((Xp0 Xx0) Xy0))
% Found (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)) as proof of ((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->((Xp0 Xx0) Xy0)))
% Found (and_rect10 (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))) as proof of ((Xp0 Xx0) Xy0)
% Found ((and_rect1 ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))) as proof of ((Xp0 Xx0) Xy0)
% Found (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)))) as proof of ((Xp0 Xx0) Xy0)
% Found (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)))) as proof of (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))
% Found (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)))) as proof of (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))
% Found (or_introl00 (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))) as proof of ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))
% Found ((or_introl0 (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))) as proof of ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))
% Found (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))) as proof of ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))
% Found (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)))))) as proof of ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))
% Found (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)))))) as proof of (((R Xx0) Xy0)->((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))))
% Found ((or_ind00 (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) as proof of ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))
% Found (((or_ind0 ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) as proof of ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))
% Found ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) as proof of ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))
% Found ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) as proof of ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))
% Found (x200 ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)))))))) as proof of ((Xp Xx0) Xy0)
% Found ((x20 Xy0) ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)))))))) as proof of ((Xp Xx0) Xy0)
% Found (((x2 Xx0) Xy0) ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)))))))) as proof of ((Xp Xx0) Xy0)
% Found (fun (x3:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> (((x2 Xx0) Xy0) ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))))) as proof of ((Xp Xx0) Xy0)
% Found (fun (Xy0:a) (x3:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> (((x2 Xx0) Xy0) ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))))) as proof of (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0))
% Found (fun (Xx0:a) (Xy0:a) (x3:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> (((x2 Xx0) Xy0) ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))))) as proof of (forall (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))
% Found (fun (Xx0:a) (Xy0:a) (x3:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> (((x2 Xx0) Xy0) ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))))) as proof of (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))
% Found (conj00 (fun (Xx0:a) (Xy0:a) (x3:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> (((x2 Xx0) Xy0) ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)))))))))) as proof of ((P Xp)->((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp)))
% Found ((conj0 (P Xp)) (fun (Xx0:a) (Xy0:a) (x3:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> (((x2 Xx0) Xy0) ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)))))))))) as proof of ((P Xp)->((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp)))
% Found (((conj (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp)) (fun (Xx0:a) (Xy0:a) (x3:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> (((x2 Xx0) Xy0) ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)))))))))) as proof of ((P Xp)->((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp)))
% Found (fun (x2:(forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((conj (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp)) (fun (Xx0:a) (Xy0:a) (x3:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> (((x2 Xx0) Xy0) ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))))))) as proof of ((P Xp)->((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp)))
% Found (fun (x2:(forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((conj (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp)) (fun (Xx0:a) (Xy0:a) (x3:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> (((x2 Xx0) Xy0) ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))))))) as proof of ((forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))->((P Xp)->((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))))
% Found (and_rect00 (fun (x2:(forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((conj (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp)) (fun (Xx0:a) (Xy0:a) (x3:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> (((x2 Xx0) Xy0) ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)))))))))))) as proof of ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))
% Found ((and_rect0 ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))) (fun (x2:(forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((conj (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp)) (fun (Xx0:a) (Xy0:a) (x3:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> (((x2 Xx0) Xy0) ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)))))))))))) as proof of ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))
% Found (((fun (P0:Type) (x2:((forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))->((P Xp)->P0)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp)) P0) x2) x0)) ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))) (fun (x2:(forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((conj (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp)) (fun (Xx0:a) (Xy0:a) (x3:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> (((x2 Xx0) Xy0) ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)))))))))))) as proof of ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))
% Found (((fun (P0:Type) (x2:((forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))->((P Xp)->P0)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp)) P0) x2) x0)) ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))) (fun (x2:(forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((conj (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp)) (fun (Xx0:a) (Xy0:a) (x3:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> (((x2 Xx0) Xy0) ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)))))))))))) as proof of ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))
% Found (x1 (((fun (P0:Type) (x2:((forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))->((P Xp)->P0)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp)) P0) x2) x0)) ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))) (fun (x2:(forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((conj (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp)) (fun (Xx0:a) (Xy0:a) (x3:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> (((x2 Xx0) Xy0) ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))))))))) as proof of ((Xp Xx) Xy)
% Found ((x Xp) (((fun (P0:Type) (x2:((forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))->((P Xp)->P0)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp)) P0) x2) x0)) ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))) (fun (x2:(forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((conj (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp)) (fun (Xx0:a) (Xy0:a) (x3:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> (((x2 Xx0) Xy0) ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))))))))) as proof of ((Xp Xx) Xy)
% Found (fun (x0:((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp)))=> ((x Xp) (((fun (P0:Type) (x2:((forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))->((P Xp)->P0)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp)) P0) x2) x0)) ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))) (fun (x2:(forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((conj (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp)) (fun (Xx0:a) (Xy0:a) (x3:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> (((x2 Xx0) Xy0) ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)))))))))))))) as proof of ((Xp Xx) Xy)
% Found (fun (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp)))=> ((x Xp) (((fun (P0:Type) (x2:((forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))->((P Xp)->P0)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp)) P0) x2) x0)) ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))) (fun (x2:(forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((conj (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp)) (fun (Xx0:a) (Xy0:a) (x3:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> (((x2 Xx0) Xy0) ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)))))))))))))) as proof of (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy))
% Found (fun (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy)))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp)))=> ((x Xp) (((fun (P0:Type) (x2:((forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))->((P Xp)->P0)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp)) P0) x2) x0)) ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))) (fun (x2:(forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((conj (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp)) (fun (Xx0:a) (Xy0:a) (x3:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> (((x2 Xx0) Xy0) ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)))))))))))))) as proof of (forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy)))
% Found (fun (Xy:a) (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy)))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp)))=> ((x Xp) (((fun (P0:Type) (x2:((forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))->((P Xp)->P0)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp)) P0) x2) x0)) ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))) (fun (x2:(forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((conj (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp)) (fun (Xx0:a) (Xy0:a) (x3:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> (((x2 Xx0) Xy0) ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)))))))))))))) as proof of ((forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy)))->(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy))))
% Found (fun (Xx:a) (Xy:a) (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy)))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp)))=> ((x Xp) (((fun (P0:Type) (x2:((forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))->((P Xp)->P0)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp)) P0) x2) x0)) ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))) (fun (x2:(forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((conj (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp)) (fun (Xx0:a) (Xy0:a) (x3:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> (((x2 Xx0) Xy0) ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)))))))))))))) as proof of (forall (Xy:a), ((forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy)))->(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy)))))
% Found (fun (S:(a->(a->Prop))) (Xx:a) (Xy:a) (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy)))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp)))=> ((x Xp) (((fun (P0:Type) (x2:((forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))->((P Xp)->P0)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp)) P0) x2) x0)) ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))) (fun (x2:(forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((conj (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp)) (fun (Xx0:a) (Xy0:a) (x3:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> (((x2 Xx0) Xy0) ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)))))))))))))) as proof of (forall (Xx:a) (Xy:a), ((forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy)))->(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy)))))
% Found (fun (R:(a->(a->Prop))) (S:(a->(a->Prop))) (Xx:a) (Xy:a) (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy)))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp)))=> ((x Xp) (((fun (P0:Type) (x2:((forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))->((P Xp)->P0)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp)) P0) x2) x0)) ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))) (fun (x2:(forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((conj (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp)) (fun (Xx0:a) (Xy0:a) (x3:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> (((x2 Xx0) Xy0) ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)))))))))))))) as proof of (forall (S:(a->(a->Prop))) (Xx:a) (Xy:a), ((forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy)))->(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy)))))
% Found (fun (P:((a->(a->Prop))->Prop)) (R:(a->(a->Prop))) (S:(a->(a->Prop))) (Xx:a) (Xy:a) (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy)))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp)))=> ((x Xp) (((fun (P0:Type) (x2:((forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))->((P Xp)->P0)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp)) P0) x2) x0)) ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))) (fun (x2:(forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((conj (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp)) (fun (Xx0:a) (Xy0:a) (x3:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> (((x2 Xx0) Xy0) ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)))))))))))))) as proof of (forall (R:(a->(a->Prop))) (S:(a->(a->Prop))) (Xx:a) (Xy:a), ((forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy)))->(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy)))))
% Found (fun (P:((a->(a->Prop))->Prop)) (R:(a->(a->Prop))) (S:(a->(a->Prop))) (Xx:a) (Xy:a) (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy)))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp)))=> ((x Xp) (((fun (P0:Type) (x2:((forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))->((P Xp)->P0)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp)) P0) x2) x0)) ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))) (fun (x2:(forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((conj (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp)) (fun (Xx0:a) (Xy0:a) (x3:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> (((x2 Xx0) Xy0) ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4)))))))))))))) as proof of (forall (P:((a->(a->Prop))->Prop)) (R:(a->(a->Prop))) (S:(a->(a->Prop))) (Xx:a) (Xy:a), ((forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy)))->(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy)))))
% Got proof (fun (P:((a->(a->Prop))->Prop)) (R:(a->(a->Prop))) (S:(a->(a->Prop))) (Xx:a) (Xy:a) (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy)))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp)))=> ((x Xp) (((fun (P0:Type) (x2:((forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))->((P Xp)->P0)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp)) P0) x2) x0)) ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))) (fun (x2:(forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((conj (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp)) (fun (Xx0:a) (Xy0:a) (x3:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> (((x2 Xx0) Xy0) ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))))))))))
% Time elapsed = 73.389332s
% node=7004 cost=1577.000000 depth=47
% ::::::::::::::::::::::
% % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p
% % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% (fun (P:((a->(a->Prop))->Prop)) (R:(a->(a->Prop))) (S:(a->(a->Prop))) (Xx:a) (Xy:a) (x:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))->((Xp Xx) Xy)))) (Xp:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp)))=> ((x Xp) (((fun (P0:Type) (x2:((forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))->((P Xp)->P0)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0)))) (P Xp)) P0) x2) x0)) ((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp))) (fun (x2:(forall (Xx0:a) (Xy0:a), (((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))->((Xp Xx0) Xy0))))=> (((conj (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp Xx0) Xy0)))) (P Xp)) (fun (Xx0:a) (Xy0:a) (x3:((or ((R Xx0) Xy0)) ((S Xx0) Xy0)))=> (((x2 Xx0) Xy0) ((((fun (P0:Prop) (x4:(((R Xx0) Xy0)->P0)) (x5:(((S Xx0) Xy0)->P0))=> ((((((or_ind ((R Xx0) Xy0)) ((S Xx0) Xy0)) P0) x4) x5) x3)) ((or (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0))))) (fun (x4:((R Xx0) Xy0))=> (((or_introl (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))) (fun (x4:((S Xx0) Xy0))=> (((or_intror (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0))->((Xp0 Xx0) Xy0)))) (fun (Xp0:(a->(a->Prop))) (x5:((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)))=> (((fun (P0:Type) (x6:((forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))->((P Xp0)->P0)))=> (((((and_rect (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (P Xp0)) P0) x6) x5)) ((Xp0 Xx0) Xy0)) (fun (x6:(forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp0 Xx1) Xy1)))) (x7:(P Xp0))=> (((x6 Xx0) Xy0) x4))))))))))))))
% % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% EOF
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