TSTP Solution File: SEV321^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV321^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n186.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 13:34:02 EDT 2014
% Result : Timeout 300.00s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEV321^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n186.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 08:50:11 CDT 2014
% % CPUTime : 300.00
% 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 0xd697e8>, <kernel.Type object at 0xd69ea8>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (R:(a->(a->Prop))) (U:((a->Prop)->a)), (((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xs:(a->Prop)), ((and (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))))->(forall (Xf:(a->a)), ((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R (Xf Xx)) (Xf Xy))))->((ex a) (fun (Xw:a)=> ((and ((R Xw) (Xf Xw))) ((R (Xf Xw)) Xw)))))))) of role conjecture named cTHM145_C_pme
% Conjecture to prove = (forall (R:(a->(a->Prop))) (U:((a->Prop)->a)), (((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xs:(a->Prop)), ((and (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))))->(forall (Xf:(a->a)), ((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R (Xf Xx)) (Xf Xy))))->((ex a) (fun (Xw:a)=> ((and ((R Xw) (Xf Xw))) ((R (Xf Xw)) Xw)))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (R:(a->(a->Prop))) (U:((a->Prop)->a)), (((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xs:(a->Prop)), ((and (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))))->(forall (Xf:(a->a)), ((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R (Xf Xx)) (Xf Xy))))->((ex a) (fun (Xw:a)=> ((and ((R Xw) (Xf Xw))) ((R (Xf Xw)) Xw))))))))']
% Parameter a:Type.
% Trying to prove (forall (R:(a->(a->Prop))) (U:((a->Prop)->a)), (((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xs:(a->Prop)), ((and (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))))->(forall (Xf:(a->a)), ((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R (Xf Xx)) (Xf Xy))))->((ex a) (fun (Xw:a)=> ((and ((R Xw) (Xf Xw))) ((R (Xf Xw)) Xw))))))))
% Found x30:=(x3 Xk):((Xs Xk)->((R Xk) (U Xs)))
% Found (x3 Xk) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x3 Xk)) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x3 Xk)) as proof of ((Xs Xk)->((Xs Xk)->((R Xk) Xy0)))
% Found (and_rect20 (fun (x7:(Xs Xk))=> (x3 Xk))) as proof of ((R Xk) Xy0)
% Found ((and_rect2 ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x3 Xk))) as proof of ((R Xk) Xy0)
% Found (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x3 Xk))) as proof of ((R Xk) Xy0)
% Found (fun (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x3 Xk)))) as proof of ((R Xk) Xy0)
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x3 Xk)))) as proof of (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0))
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x3 Xk)))) as proof of (forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0)))
% Found (x40 (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x3 Xk))))) as proof of ((and ((R x5) Xy0)) ((R Xy0) Xy))
% Found ((x4 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x3 Xk))))) as proof of ((and ((R x5) Xy0)) ((R Xy0) Xy))
% Found ((x4 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x3 Xk))))) as proof of ((and ((R x5) Xy0)) ((R Xy0) Xy))
% Found ((x4 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x3 Xk))))) as proof of ((and ((R x5) Xy0)) ((R Xy0) Xy))
% Found ((x4 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x3 Xk))))) as proof of ((and ((R x5) Xy0)) ((R Xy0) Xy))
% Found ((x4 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x3 Xk))))) as proof of ((and ((R x5) Xy0)) ((R Xy0) Xy))
% Found (x1010 ((x4 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x3 Xk)))))) as proof of ((R x5) Xy)
% Found ((x101 (U Xs)) ((x4 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x3 Xk)))))) as proof of ((R x5) Xy)
% Found (((fun (Xy0:a)=> ((x10 Xy0) Xy)) (U Xs)) ((x4 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x3 Xk)))))) as proof of ((R x5) Xy)
% Found (((fun (Xy0:a)=> ((x10 Xy0) Xy)) (U Xs)) ((x4 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x3 Xk)))))) as proof of ((R x5) Xy)
% Found x40:=(x4 Xk):((Xs Xk)->((R Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x4 Xk)) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x4 Xk)) as proof of ((Xs Xk)->((Xs Xk)->((R Xk) Xy0)))
% Found (and_rect20 (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found ((and_rect2 ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found (fun (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of ((R Xk) Xy0)
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0))
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of (forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0)))
% Found (x50 (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found (x1010 ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x3) Xy)
% Found ((x101 (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x3) Xy)
% Found (((fun (Xy0:a)=> ((x10 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x3) Xy)
% Found (((fun (Xy0:a)=> ((x10 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x3) Xy)
% Found x40:=(x4 Xk):((Xs Xk)->((R Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x4 Xk)) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x4 Xk)) as proof of ((Xs Xk)->((Xs Xk)->((R Xk) Xy0)))
% Found (and_rect20 (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found ((and_rect2 ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found (fun (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of ((R Xk) Xy0)
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0))
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of (forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0)))
% Found (x50 (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found (x2010 ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x1) Xy)
% Found ((x201 (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x1) Xy)
% Found (((fun (Xy0:a)=> ((x20 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x1) Xy)
% Found (((fun (Xy0:a)=> ((x20 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x1) Xy)
% Found x40:=(x4 Xk):((Xs Xk)->((R Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x4 Xk)) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x4 Xk)) as proof of ((Xs Xk)->((Xs Xk)->((R Xk) Xy0)))
% Found (and_rect20 (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found ((and_rect2 ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found (fun (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of ((R Xk) Xy0)
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0))
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of (forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0)))
% Found (x50 (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found (x1010 ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x3) Xy)
% Found ((x101 (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x3) Xy)
% Found (((fun (Xy0:a)=> ((x10 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x3) Xy)
% Found (((fun (Xy0:a)=> ((x10 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x3) Xy)
% Found x40:=(x4 Xk):((Xs Xk)->((R Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x4 Xk)) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x4 Xk)) as proof of ((Xs Xk)->((Xs Xk)->((R Xk) Xy0)))
% Found (and_rect20 (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found ((and_rect2 ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found (fun (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of ((R Xk) Xy0)
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0))
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of (forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0)))
% Found (x50 (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found (x1010 ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x3) Xy)
% Found ((x101 (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x3) Xy)
% Found (((fun (Xy0:a)=> ((x10 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x3) Xy)
% Found (((fun (Xy0:a)=> ((x10 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x3) Xy)
% Found x40:=(x4 Xk):((Xs Xk)->((R Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x4 Xk)) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x4 Xk)) as proof of ((Xs Xk)->((Xs Xk)->((R Xk) Xy0)))
% Found (and_rect20 (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found ((and_rect2 ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found (fun (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of ((R Xk) Xy0)
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0))
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of (forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0)))
% Found (x50 (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found (x2010 ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x1) Xy)
% Found ((x201 (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x1) Xy)
% Found (((fun (Xy0:a)=> ((x20 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x1) Xy)
% Found (((fun (Xy0:a)=> ((x20 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x1) Xy)
% Found x40:=(x4 Xk):((Xs Xk)->((R Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x4 Xk)) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x4 Xk)) as proof of ((Xs Xk)->((Xs Xk)->((R Xk) Xy0)))
% Found (and_rect20 (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found ((and_rect2 ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found (fun (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of ((R Xk) Xy0)
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0))
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of (forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0)))
% Found (x50 (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found (x1010 ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x3) Xy)
% Found ((x101 (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x3) Xy)
% Found (((fun (Xy0:a)=> ((x10 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x3) Xy)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> (((fun (Xy0:a)=> ((x10 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk))))))) as proof of ((R x3) Xy)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> (((fun (Xy0:a)=> ((x10 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk))))))) as proof of ((forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj))))->((R x3) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> (((fun (Xy0:a)=> ((x10 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk))))))) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj))))->((R x3) Xy)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> (((fun (Xy0:a)=> ((x10 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))))) as proof of ((R x3) Xy)
% Found ((and_rect1 ((R x3) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> (((fun (Xy0:a)=> ((x10 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))))) as proof of ((R x3) Xy)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj))))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj))))) P) x4) x20)) ((R x3) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> (((fun (Xy0:a)=> ((x10 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))))) as proof of ((R x3) Xy)
% Found x40:=(x4 Xk):((Xs Xk)->((R Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x4 Xk)) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x4 Xk)) as proof of ((Xs Xk)->((Xs Xk)->((R Xk) Xy0)))
% Found (and_rect20 (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found ((and_rect2 ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found (fun (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of ((R Xk) Xy0)
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0))
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of (forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0)))
% Found (x50 (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found (x2010 ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x1) Xy)
% Found ((x201 (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x1) Xy)
% Found (((fun (Xy0:a)=> ((x20 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x1) Xy)
% Found (((fun (Xy0:a)=> ((x20 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x1) Xy)
% Found x40:=(x4 Xk):((Xs Xk)->((R Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x4 Xk)) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x4 Xk)) as proof of ((Xs Xk)->((Xs Xk)->((R Xk) Xy0)))
% Found (and_rect20 (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found ((and_rect2 ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found (fun (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of ((R Xk) Xy0)
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0))
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of (forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0)))
% Found (x50 (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found (x2010 ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x1) Xy)
% Found ((x201 (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x1) Xy)
% Found (((fun (Xy0:a)=> ((x20 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x1) Xy)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> (((fun (Xy0:a)=> ((x20 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk))))))) as proof of ((R x1) Xy)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> (((fun (Xy0:a)=> ((x20 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk))))))) as proof of ((forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj))))->((R x1) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> (((fun (Xy0:a)=> ((x20 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk))))))) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj))))->((R x1) Xy)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> (((fun (Xy0:a)=> ((x20 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))))) as proof of ((R x1) Xy)
% Found ((and_rect1 ((R x1) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> (((fun (Xy0:a)=> ((x20 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))))) as proof of ((R x1) Xy)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj))))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj))))) P) x4) x30)) ((R x1) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> (((fun (Xy0:a)=> ((x20 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))))) as proof of ((R x1) Xy)
% Found x40:=(x4 Xk):((Xs Xk)->((R Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x4 Xk)) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x4 Xk)) as proof of ((Xs Xk)->((Xs Xk)->((R Xk) Xy0)))
% Found (and_rect20 (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found ((and_rect2 ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found (fun (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of ((R Xk) Xy0)
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0))
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of (forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0)))
% Found (x50 (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found (x2010 ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x1) Xy)
% Found ((x201 (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x1) Xy)
% Found (((fun (Xy0:a)=> ((x20 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x1) Xy)
% Found (((fun (Xy0:a)=> ((x20 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x1) Xy)
% Found x40:=(x4 Xk):((Xs Xk)->((R Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x4 Xk)) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x4 Xk)) as proof of ((Xs Xk)->((Xs Xk)->((R Xk) Xy0)))
% Found (and_rect20 (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found ((and_rect2 ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found (fun (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of ((R Xk) Xy0)
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0))
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of (forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0)))
% Found (x50 (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found (x2010 ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x1) Xy)
% Found ((x201 (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x1) Xy)
% Found (((fun (Xy0:a)=> ((x20 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x1) Xy)
% Found (((fun (Xy0:a)=> ((x20 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x1) Xy)
% Found x40:=(x4 Xk):((Xs Xk)->((R Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x4 Xk)) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x4 Xk)) as proof of ((Xs Xk)->((Xs Xk)->((R Xk) Xy0)))
% Found (and_rect20 (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found ((and_rect2 ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found (fun (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of ((R Xk) Xy0)
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0))
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of (forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0)))
% Found (x50 (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found (x2010 ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x1) Xy)
% Found ((x201 (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x1) Xy)
% Found (((fun (Xy0:a)=> ((x20 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((R x1) Xy)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> (((fun (Xy0:a)=> ((x20 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk))))))) as proof of ((R x1) Xy)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> (((fun (Xy0:a)=> ((x20 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk))))))) as proof of ((forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj))))->((R x1) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> (((fun (Xy0:a)=> ((x20 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk))))))) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj))))->((R x1) Xy)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> (((fun (Xy0:a)=> ((x20 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))))) as proof of ((R x1) Xy)
% Found ((and_rect1 ((R x1) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> (((fun (Xy0:a)=> ((x20 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))))) as proof of ((R x1) Xy)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj))))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj))))) P) x4) x30)) ((R x1) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> (((fun (Xy0:a)=> ((x20 Xy0) Xy)) (U Xs)) ((x5 (U Xs)) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) (U Xs))) (fun (x7:(Xs Xk))=> (x4 Xk)))))))) as proof of ((R x1) Xy)
% Found x40:=(x4 Xk):((Xs Xk)->((R Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x4 Xk)) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x4 Xk)) as proof of ((Xs Xk)->((Xs Xk)->((R Xk) Xy0)))
% Found (and_rect20 (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found ((and_rect2 ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found (fun (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of ((R Xk) Xy0)
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0))
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of (forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0)))
% Found (x50 (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj))))->((and ((R x3) Xy0)) ((R Xy0) Xy)))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj))))->((and ((R x3) Xy0)) ((R Xy0) Xy))))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found ((and_rect1 ((and ((R x3) Xy0)) ((R Xy0) Xy))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj))))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj))))) P) x4) x20)) ((and ((R x3) Xy0)) ((R Xy0) Xy))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found x40:=(x4 Xk):((Xs Xk)->((R Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x4 Xk)) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x4 Xk)) as proof of ((Xs Xk)->((Xs Xk)->((R Xk) Xy0)))
% Found (and_rect20 (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found ((and_rect2 ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found (fun (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of ((R Xk) Xy0)
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0))
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of (forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0)))
% Found (x50 (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj))))->((and ((R x1) Xy0)) ((R Xy0) Xy)))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))))) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj))))->((and ((R x1) Xy0)) ((R Xy0) Xy))))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((and_rect1 ((and ((R x1) Xy0)) ((R Xy0) Xy))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj))))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj))))) P) x4) x30)) ((and ((R x1) Xy0)) ((R Xy0) Xy))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xj)))->((and ((R (U Xs)) Xj)) ((R (U Xs)) Xj)))))=> ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found x40:=(x4 Xk):((Xs Xk)->((R Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x4 Xk)) as proof of ((Xs Xk)->((R Xk) Xy0))
% Found (fun (x7:(Xs Xk))=> (x4 Xk)) as proof of ((Xs Xk)->((Xs Xk)->((R Xk) Xy0)))
% Found (and_rect20 (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found ((and_rect2 ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))) as proof of ((R Xk) Xy0)
% Found (fun (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of ((R Xk) Xy0)
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0))
% Found (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk)))) as proof of (forall (Xk:a), (((and (Xs Xk)) (Xs Xk))->((R Xk) Xy0)))
% Found (x50 (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((x5 Xy0) (fun (Xk:a) (x6:((and (Xs Xk)) (Xs Xk)))=> (((fun (P:Type) (x7:((Xs Xk)->((Xs Xk)->P)))=> (((((and_rect (Xs Xk)) (Xs Xk)) P) x7) x6)) ((R Xk) Xy0)) (fun (x7:(Xs Xk))=> (x4 Xk))))) as proof of ((and ((R x1)
% EOF
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