%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV319^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n111.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 300.02s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV319^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n111.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 08:49:46 CDT 2014
% % CPUTime  : 300.02 
% 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 0x155e3f8>, <kernel.Type object at 0x155e1b8>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (L:(a->(a->Prop))) (U:((a->Prop)->a)), (((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((L Xx) Xy)) ((L Xy) Xz))->((L Xx) Xz)))) (forall (Xs:(a->Prop)), ((and (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))))->(forall (Xf:(a->a)), ((forall (Xx:a) (Xy:a), (((L Xx) Xy)->((L (Xf Xx)) (Xf Xy))))->((ex a) (fun (Xw:a)=> ((and ((L Xw) (Xf Xw))) ((L (Xf Xw)) Xw)))))))) of role conjecture named cTHM145L_pme
% Conjecture to prove = (forall (L:(a->(a->Prop))) (U:((a->Prop)->a)), (((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((L Xx) Xy)) ((L Xy) Xz))->((L Xx) Xz)))) (forall (Xs:(a->Prop)), ((and (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))))->(forall (Xf:(a->a)), ((forall (Xx:a) (Xy:a), (((L Xx) Xy)->((L (Xf Xx)) (Xf Xy))))->((ex a) (fun (Xw:a)=> ((and ((L Xw) (Xf Xw))) ((L (Xf Xw)) Xw)))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (L:(a->(a->Prop))) (U:((a->Prop)->a)), (((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((L Xx) Xy)) ((L Xy) Xz))->((L Xx) Xz)))) (forall (Xs:(a->Prop)), ((and (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))))->(forall (Xf:(a->a)), ((forall (Xx:a) (Xy:a), (((L Xx) Xy)->((L (Xf Xx)) (Xf Xy))))->((ex a) (fun (Xw:a)=> ((and ((L Xw) (Xf Xw))) ((L (Xf Xw)) Xw))))))))']
% Parameter a:Type.
% Trying to prove (forall (L:(a->(a->Prop))) (U:((a->Prop)->a)), (((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((L Xx) Xy)) ((L Xy) Xz))->((L Xx) Xz)))) (forall (Xs:(a->Prop)), ((and (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))))->(forall (Xf:(a->a)), ((forall (Xx:a) (Xy:a), (((L Xx) Xy)->((L (Xf Xx)) (Xf Xy))))->((ex a) (fun (Xw:a)=> ((and ((L Xw) (Xf Xw))) ((L (Xf Xw)) Xw))))))))
% Found x3:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: x5:=(U Xs):a
% Found x3 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x5)))
% Found (x40 x3) as proof of ((L Xy) x5)
% Found ((x4 x5) x3) as proof of ((L Xy) x5)
% Found ((x4 x5) x3) as proof of ((L Xy) x5)
% Found ((x4 x5) x3) as proof of ((L Xy) x5)
% Found x3:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x3 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x40 x3) as proof of ((L x5) Xy)
% Found ((x4 Xy) x3) as proof of ((L x5) Xy)
% Found ((x4 Xy) x3) as proof of ((L x5) Xy)
% Found ((x4 Xy) x3) as proof of ((L x5) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((L Xk) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: x3:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x3)))
% Found (x50 x4) as proof of ((L Xy) x3)
% Found ((x5 x3) x4) as proof of ((L Xy) x3)
% Found ((x5 x3) x4) as proof of ((L Xy) x3)
% Found ((x5 x3) x4) as proof of ((L Xy) x3)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L x3) Xy)
% Found ((x5 Xy) x4) as proof of ((L x3) Xy)
% Found ((x5 Xy) x4) as proof of ((L x3) Xy)
% Found ((x5 Xy) x4) as proof of ((L x3) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((L Xk) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: x1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x1)))
% Found (x50 x4) as proof of ((L Xy) x1)
% Found ((x5 x1) x4) as proof of ((L Xy) x1)
% Found ((x5 x1) x4) as proof of ((L Xy) x1)
% Found ((x5 x1) x4) as proof of ((L Xy) x1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L x1) Xy)
% Found ((x5 Xy) x4) as proof of ((L x1) Xy)
% Found ((x5 Xy) x4) as proof of ((L x1) Xy)
% Found ((x5 Xy) x4) as proof of ((L x1) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((L Xk) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: x3:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x3)))
% Found (x50 x4) as proof of ((L Xy) x3)
% Found ((x5 x3) x4) as proof of ((L Xy) x3)
% Found ((x5 x3) x4) as proof of ((L Xy) x3)
% Found ((x5 x3) x4) as proof of ((L Xy) x3)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L x3) Xy)
% Found ((x5 Xy) x4) as proof of ((L x3) Xy)
% Found ((x5 Xy) x4) as proof of ((L x3) Xy)
% Found ((x5 Xy) x4) as proof of ((L x3) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((L Xk) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: x3:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x3)))
% Found (x50 x4) as proof of ((L Xy) x3)
% Found ((x5 x3) x4) as proof of ((L Xy) x3)
% Found ((x5 x3) x4) as proof of ((L Xy) x3)
% Found ((x5 x3) x4) as proof of ((L Xy) x3)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: x1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x1)))
% Found (x50 x4) as proof of ((L Xy) x1)
% Found ((x5 x1) x4) as proof of ((L Xy) x1)
% Found ((x5 x1) x4) as proof of ((L Xy) x1)
% Found ((x5 x1) x4) as proof of ((L Xy) x1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L x3) Xy)
% Found ((x5 Xy) x4) as proof of ((L x3) Xy)
% Found ((x5 Xy) x4) as proof of ((L x3) Xy)
% Found ((x5 Xy) x4) as proof of ((L x3) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L x1) Xy)
% Found ((x5 Xy) x4) as proof of ((L x1) Xy)
% Found ((x5 Xy) x4) as proof of ((L x1) Xy)
% Found ((x5 Xy) x4) as proof of ((L x1) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((L Xk) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: x1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x1)))
% Found (x50 x4) as proof of ((L Xy) x1)
% Found ((x5 x1) x4) as proof of ((L Xy) x1)
% Found ((x5 x1) x4) as proof of ((L Xy) x1)
% Found ((x5 x1) x4) as proof of ((L Xy) x1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L x1) Xy)
% Found ((x5 Xy) x4) as proof of ((L x1) Xy)
% Found ((x5 Xy) x4) as proof of ((L x1) Xy)
% Found ((x5 Xy) x4) as proof of ((L x1) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: x1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x1)))
% Found (x50 x4) as proof of ((L Xy) x1)
% Found ((x5 x1) x4) as proof of ((L Xy) x1)
% Found ((x5 x1) x4) as proof of ((L Xy) x1)
% Found ((x5 x1) x4) as proof of ((L Xy) x1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L x1) Xy)
% Found ((x5 Xy) x4) as proof of ((L x1) Xy)
% Found ((x5 Xy) x4) as proof of ((L x1) Xy)
% Found ((x5 Xy) x4) as proof of ((L x1) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((L Xk) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: x3:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x3)))
% Found (x50 x4) as proof of ((L Xy) x3)
% Found ((x5 x3) x4) as proof of ((L Xy) x3)
% Found ((x5 x3) x4) as proof of ((L Xy) x3)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x3) x4)) as proof of ((L Xy) x3)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x3) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy) x3))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x3) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy) x3)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x3) x4))) as proof of ((L Xy) x3)
% Found ((and_rect1 ((L Xy) x3)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x3) x4))) as proof of ((L Xy) x3)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x20)) ((L Xy) x3)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x3) x4))) as proof of ((L Xy) x3)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L x3) Xy)
% Found ((x5 Xy) x4) as proof of ((L x3) Xy)
% Found ((x5 Xy) x4) as proof of ((L x3) Xy)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((L x3) Xy)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L x3) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L x3) Xy)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((L x3) Xy)
% Found ((and_rect1 ((L x3) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((L x3) Xy)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x20)) ((L x3) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((L x3) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: x1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x1)))
% Found (x50 x4) as proof of ((L Xy) x1)
% Found ((x5 x1) x4) as proof of ((L Xy) x1)
% Found ((x5 x1) x4) as proof of ((L Xy) x1)
% Found ((x5 x1) x4) as proof of ((L Xy) x1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L x1) Xy)
% Found ((x5 Xy) x4) as proof of ((L x1) Xy)
% Found ((x5 Xy) x4) as proof of ((L x1) Xy)
% Found ((x5 Xy) x4) as proof of ((L x1) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: x1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x1)))
% Found (x50 x4) as proof of ((L Xy) x1)
% Found ((x5 x1) x4) as proof of ((L Xy) x1)
% Found ((x5 x1) x4) as proof of ((L Xy) x1)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) x4)) as proof of ((L Xy) x1)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy) x1))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy) x1)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) x4))) as proof of ((L Xy) x1)
% Found ((and_rect1 ((L Xy) x1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) x4))) as proof of ((L Xy) x1)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x30)) ((L Xy) x1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) x4))) as proof of ((L Xy) x1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L x1) Xy)
% Found ((x5 Xy) x4) as proof of ((L x1) Xy)
% Found ((x5 Xy) x4) as proof of ((L x1) Xy)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((L x1) Xy)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L x1) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L x1) Xy)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((L x1) Xy)
% Found ((and_rect1 ((L x1) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((L x1) Xy)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x30)) ((L x1) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((L x1) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: x1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x1)))
% Found (x50 x4) as proof of ((L Xy) x1)
% Found ((x5 x1) x4) as proof of ((L Xy) x1)
% Found ((x5 x1) x4) as proof of ((L Xy) x1)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) x4)) as proof of ((L Xy) x1)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy) x1))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy) x1)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) x4))) as proof of ((L Xy) x1)
% Found ((and_rect1 ((L Xy) x1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) x4))) as proof of ((L Xy) x1)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x30)) ((L Xy) x1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) x4))) as proof of ((L Xy) x1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L x1) Xy)
% Found ((x5 Xy) x4) as proof of ((L x1) Xy)
% Found ((x5 Xy) x4) as proof of ((L x1) Xy)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((L x1) Xy)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L x1) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L x1) Xy)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((L x1) Xy)
% Found ((and_rect1 ((L x1) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((L x1) Xy)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x30)) ((L x1) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((L x1) Xy)
% Found x3:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xs0:=Xs:(a->Prop);x7:=(U Xs):a
% Found x3 as proof of (forall (Xk:a), ((Xs0 Xk)->((L Xk) x7)))
% Found (x60 x3) as proof of ((L Xy) x7)
% Found ((x6 x7) x3) as proof of ((L Xy) x7)
% Found ((x6 x7) x3) as proof of ((L Xy) x7)
% Found ((x6 x7) x3) as proof of ((L Xy) x7)
% Found x3:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xs0:=Xs:(a->Prop);Xy:=(U Xs):a
% Found x3 as proof of (forall (Xk:a), ((Xs0 Xk)->((L Xk) Xy)))
% Found (x60 x3) as proof of ((L x7) Xy)
% Found ((x6 Xy) x3) as proof of ((L x7) Xy)
% Found ((x6 Xy) x3) as proof of ((L x7) Xy)
% Found ((x6 Xy) x3) as proof of ((L x7) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x3:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xs0:=Xs:(a->Prop);x5:=(U Xs):a
% Found x3 as proof of (forall (Xk:a), ((Xs0 Xk)->((L Xk) x5)))
% Found (x70 x3) as proof of ((L Xy) x5)
% Found ((x7 x5) x3) as proof of ((L Xy) x5)
% Found ((x7 x5) x3) as proof of ((L Xy) x5)
% Found ((x7 x5) x3) as proof of ((L Xy) x5)
% Found x3:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x3 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x40 x3) as proof of ((L x5) Xy)
% Found ((x4 Xy) x3) as proof of ((L x5) Xy)
% Found ((x4 Xy) x3) as proof of ((L x5) Xy)
% Found ((x4 Xy) x3) as proof of ((L x5) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: x3:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((L Xk) x3)))
% Found (x70 x6) as proof of ((L Xy) x3)
% Found ((x7 x3) x6) as proof of ((L Xy) x3)
% Found ((x7 x3) x6) as proof of ((L Xy) x3)
% Found ((x7 x3) x6) as proof of ((L Xy) x3)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((L Xk) Xy)))
% Found (x70 x6) as proof of ((L x3) Xy)
% Found ((x7 Xy) x6) as proof of ((L x3) Xy)
% Found ((x7 Xy) x6) as proof of ((L x3) Xy)
% Found ((x7 Xy) x6) as proof of ((L x3) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: x1:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((L Xk) x1)))
% Found (x70 x6) as proof of ((L Xy) x1)
% Found ((x7 x1) x6) as proof of ((L Xy) x1)
% Found ((x7 x1) x6) as proof of ((L Xy) x1)
% Found ((x7 x1) x6) as proof of ((L Xy) x1)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xs:=Xs0:(a->Prop);Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x6) as proof of ((L x1) Xy)
% Found ((x5 Xy) x6) as proof of ((L x1) Xy)
% Found ((x5 Xy) x6) as proof of ((L x1) Xy)
% Found ((x5 Xy) x6) as proof of ((L x1) Xy)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x3:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x3 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x40 x3) as proof of ((L x5) Xy0)
% Found ((x4 Xy0) x3) as proof of ((L x5) Xy0)
% Found ((x4 Xy0) x3) as proof of ((L x5) Xy0)
% Found x400:=(x40 x3):((L (U Xs)) x5)
% Found (x40 x3) as proof of ((L Xy0) x5)
% Found ((x4 x5) x3) as proof of ((L Xy0) x5)
% Found ((x4 x5) x3) as proof of ((L Xy0) x5)
% Found ((x4 x5) x3) as proof of ((L Xy0) x5)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found x500:=(x50 x4):((L (U Xs)) x3)
% Found (x50 x4) as proof of ((L Xy0) x3)
% Found ((x5 x3) x4) as proof of ((L Xy0) x3)
% Found ((x5 x3) x4) as proof of ((L Xy0) x3)
% Found ((x5 x3) x4) as proof of ((L Xy0) x3)
% Found x3:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: x5:=(U Xs):a
% Found x3 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x5)))
% Found (x40 x3) as proof of ((L Xy) x5)
% Found ((x4 x5) x3) as proof of ((L Xy) x5)
% Found ((x4 x5) x3) as proof of ((L Xy) x5)
% Found ((x4 x5) x3) as proof of ((L Xy) x5)
% Found x3:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xs0:=Xs:(a->Prop);Xy:=(U Xs):a
% Found x3 as proof of (forall (Xk:a), ((Xs0 Xk)->((L Xk) Xy)))
% Found (x70 x3) as proof of ((L x5) Xy)
% Found ((x7 Xy) x3) as proof of ((L x5) Xy)
% Found ((x7 Xy) x3) as proof of ((L x5) Xy)
% Found ((x7 Xy) x3) as proof of ((L x5) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(L x5):(a->Prop)
% Found x6 as proof of ((L x5) Xy0)
% Found x500:=(x50 x4):((L (U Xs)) x1)
% Found (x50 x4) as proof of ((L Xy0) x1)
% Found ((x5 x1) x4) as proof of ((L Xy0) x1)
% Found ((x5 x1) x4) as proof of ((L Xy0) x1)
% Found ((x5 x1) x4) as proof of ((L Xy0) x1)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x3:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xs0:=Xs:(a->Prop);x5:=(U Xs):a
% Found x3 as proof of (forall (Xk:a), ((Xs0 Xk)->((L Xk) x5)))
% Found (x70 x3) as proof of ((L Xy) x5)
% Found ((x7 x5) x3) as proof of ((L Xy) x5)
% Found ((x7 x5) x3) as proof of ((L Xy) x5)
% Found ((x7 x5) x3) as proof of ((L Xy) x5)
% Found x3:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x3 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x40 x3) as proof of ((L x5) Xy)
% Found ((x4 Xy) x3) as proof of ((L x5) Xy)
% Found ((x4 Xy) x3) as proof of ((L x5) Xy)
% Found ((x4 Xy) x3) as proof of ((L x5) Xy)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: x3:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((L Xk) x3)))
% Found (x70 x6) as proof of ((L Xy) x3)
% Found ((x7 x3) x6) as proof of ((L Xy) x3)
% Found ((x7 x3) x6) as proof of ((L Xy) x3)
% Found ((x7 x3) x6) as proof of ((L Xy) x3)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((L Xk) Xy)))
% Found (x70 x6) as proof of ((L x3) Xy)
% Found ((x7 Xy) x6) as proof of ((L x3) Xy)
% Found ((x7 Xy) x6) as proof of ((L x3) Xy)
% Found ((x7 Xy) x6) as proof of ((L x3) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(L x3):(a->Prop)
% Found x6 as proof of ((L x3) Xy0)
% Found x3:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x3 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x40 x3) as proof of ((L x5) Xy0)
% Found ((x4 Xy0) x3) as proof of ((L x5) Xy0)
% Found ((x4 Xy0) x3) as proof of ((L x5) Xy0)
% Found ((x4 Xy0) x3) as proof of ((L x5) Xy0)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((L Xk) Xy)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xs:=Xs0:(a->Prop);x3:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x3)))
% Found (x50 x6) as proof of ((L Xy) x3)
% Found ((x5 x3) x6) as proof of ((L Xy) x3)
% Found ((x5 x3) x6) as proof of ((L Xy) x3)
% Found ((x5 x3) x6) as proof of ((L Xy) x3)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xs:=Xs0:(a->Prop);Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x6) as proof of ((L x3) Xy)
% Found ((x5 Xy) x6) as proof of ((L x3) Xy)
% Found ((x5 Xy) x6) as proof of ((L x3) Xy)
% Found ((x5 Xy) x6) as proof of ((L x3) Xy)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: x1:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((L Xk) x1)))
% Found (x70 x6) as proof of ((L Xy) x1)
% Found ((x7 x1) x6) as proof of ((L Xy) x1)
% Found ((x7 x1) x6) as proof of ((L Xy) x1)
% Found ((x7 x1) x6) as proof of ((L Xy) x1)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((L Xk) Xy)))
% Found (x70 x6) as proof of ((L x1) Xy)
% Found ((x7 Xy) x6) as proof of ((L x1) Xy)
% Found ((x7 Xy) x6) as proof of ((L x1) Xy)
% Found ((x7 Xy) x6) as proof of ((L x1) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(L x1):(a->Prop)
% Found x6 as proof of ((L x1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((L Xk) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((L Xk) Xy)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: x1:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((L Xk) x1)))
% Found (x70 x6) as proof of ((L Xy) x1)
% Found ((x7 x1) x6) as proof of ((L Xy) x1)
% Found ((x7 x1) x6) as proof of ((L Xy) x1)
% Found ((x7 x1) x6) as proof of ((L Xy) x1)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((L Xk) Xy)))
% Found (x70 x6) as proof of ((L x1) Xy)
% Found ((x7 Xy) x6) as proof of ((L x1) Xy)
% Found ((x7 Xy) x6) as proof of ((L x1) Xy)
% Found ((x7 Xy) x6) as proof of ((L x1) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a
% Found x6 as proof of ((L Xk) Xy0)
% Found x3:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x3 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x40 x3) as proof of ((L Xy0) Xy)
% Found ((x4 Xy) x3) as proof of ((L Xy0) Xy)
% Found ((x4 Xy) x3) as proof of ((L Xy0) Xy)
% Found ((x4 Xy) x3) as proof of ((L Xy0) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((L Xk) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((L Xk) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a
% Found x6 as proof of ((L Xk) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(L Xk):(a->Prop)
% Found x6 as proof of ((L Xk) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found x500:=(x50 x4):((L (U Xs)) x3)
% Found (x50 x4) as proof of ((L Xy0) x3)
% Found ((x5 x3) x4) as proof of ((L Xy0) x3)
% Found ((x5 x3) x4) as proof of ((L Xy0) x3)
% Found ((x5 x3) x4) as proof of ((L Xy0) x3)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((L Xk) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a
% Found x6 as proof of ((L Xk) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(L Xk):(a->Prop)
% Found x6 as proof of ((L Xk) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found x500:=(x50 x4):((L (U Xs)) x3)
% Found (x50 x4) as proof of ((L Xy0) x3)
% Found ((x5 x3) x4) as proof of ((L Xy0) x3)
% Found ((x5 x3) x4) as proof of ((L Xy0) x3)
% Found ((x5 x3) x4) as proof of ((L Xy0) x3)
% Found x500:=(x50 x4):((L (U Xs)) x1)
% Found (x50 x4) as proof of ((L Xy0) x1)
% Found ((x5 x1) x4) as proof of ((L Xy0) x1)
% Found ((x5 x1) x4) as proof of ((L Xy0) x1)
% Found ((x5 x1) x4) as proof of ((L Xy0) x1)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xs:=Xs0:(a->Prop);x3:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x3)))
% Found (x50 x6) as proof of ((L Xy) x3)
% Found ((x5 x3) x6) as proof of ((L Xy) x3)
% Found ((x5 x3) x6) as proof of ((L Xy) x3)
% Found ((x5 x3) x6) as proof of ((L Xy) x3)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xs:=Xs0:(a->Prop);Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x6) as proof of ((L x3) Xy)
% Found ((x5 Xy) x6) as proof of ((L x3) Xy)
% Found ((x5 Xy) x6) as proof of ((L x3) Xy)
% Found ((x5 Xy) x6) as proof of ((L x3) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found x30:=(x3 Xk):((Xs Xk)->((L Xk) (U Xs)))
% Found (x3 Xk) as proof of ((Xs Xk)->((L Xk) x5))
% Found (x3 Xk) as proof of ((Xs Xk)->((L Xk) x5))
% Found (fun (Xk:a)=> (x3 Xk)) as proof of ((Xs Xk)->((L Xk) x5))
% Found (fun (Xk:a)=> (x3 Xk)) as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x5)))
% Found (x40 (fun (Xk:a)=> (x3 Xk))) as proof of ((L Xy0) x5)
% Found ((x4 x5) (fun (Xk:a)=> (x3 Xk))) as proof of ((L Xy0) x5)
% Found ((x4 x5) (fun (Xk:a)=> (x3 Xk))) as proof of ((L Xy0) x5)
% Found ((x4 x5) (fun (Xk:a)=> (x3 Xk))) as proof of ((L Xy0) x5)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(L Xk):(a->Prop)
% Found x6 as proof of ((L Xk) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(L x3):(a->Prop)
% Found x6 as proof of ((L x3) Xy0)
% Found x500:=(x50 x4):((L (U Xs)) x1)
% Found (x50 x4) as proof of ((L Xy0) x1)
% Found ((x5 x1) x4) as proof of ((L Xy0) x1)
% Found ((x5 x1) x4) as proof of ((L Xy0) x1)
% Found ((x5 x1) x4) as proof of ((L Xy0) x1)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: x3:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((L Xk) x3)))
% Found (x70 x6) as proof of ((L Xy) x3)
% Found ((x7 x3) x6) as proof of ((L Xy) x3)
% Found ((x7 x3) x6) as proof of ((L Xy) x3)
% Found ((x7 x3) x6) as proof of ((L Xy) x3)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((L Xk) Xy)))
% Found (x70 x6) as proof of ((L x3) Xy)
% Found ((x7 Xy) x6) as proof of ((L x3) Xy)
% Found ((x7 Xy) x6) as proof of ((L x3) Xy)
% Found ((x7 Xy) x6) as proof of ((L x3) Xy)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xs:=Xs0:(a->Prop);x1:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x1)))
% Found (x50 x6) as proof of ((L Xy) x1)
% Found ((x5 x1) x6) as proof of ((L Xy) x1)
% Found ((x5 x1) x6) as proof of ((L Xy) x1)
% Found ((x5 x1) x6) as proof of ((L Xy) x1)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xs:=Xs0:(a->Prop);Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x6) as proof of ((L x1) Xy)
% Found ((x5 Xy) x6) as proof of ((L x1) Xy)
% Found ((x5 Xy) x6) as proof of ((L x1) Xy)
% Found ((x5 Xy) x6) as proof of ((L x1) Xy)
% Found x40:=(x4 Xk):((Xs Xk)->((L Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x3))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x3))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of ((Xs Xk)->((L Xk) x3))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x3)))
% Found (x50 (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x3)
% Found ((x5 x3) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x3)
% Found ((x5 x3) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x3)
% Found ((x5 x3) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x3)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(L x3):(a->Prop)
% Found x6 as proof of ((L x3) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(L x1):(a->Prop)
% Found x6 as proof of ((L x1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((L Xk) Xy)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xs:=Xs0:(a->Prop);x3:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x3)))
% Found (x50 x6) as proof of ((L Xy) x3)
% Found ((x5 x3) x6) as proof of ((L Xy) x3)
% Found ((x5 x3) x6) as proof of ((L Xy) x3)
% Found ((x5 x3) x6) as proof of ((L Xy) x3)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: x1:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((L Xk) x1)))
% Found (x70 x6) as proof of ((L Xy) x1)
% Found ((x7 x1) x6) as proof of ((L Xy) x1)
% Found ((x7 x1) x6) as proof of ((L Xy) x1)
% Found ((x7 x1) x6) as proof of ((L Xy) x1)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xs:=Xs0:(a->Prop);Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x6) as proof of ((L x3) Xy)
% Found ((x5 Xy) x6) as proof of ((L x3) Xy)
% Found ((x5 Xy) x6) as proof of ((L x3) Xy)
% Found ((x5 Xy) x6) as proof of ((L x3) Xy)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((L Xk) Xy)))
% Found (x70 x6) as proof of ((L x1) Xy)
% Found ((x7 Xy) x6) as proof of ((L x1) Xy)
% Found ((x7 Xy) x6) as proof of ((L x1) Xy)
% Found ((x7 Xy) x6) as proof of ((L x1) Xy)
% Found x40:=(x4 Xk):((Xs Xk)->((L Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x1))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x1))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of ((Xs Xk)->((L Xk) x1))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x1)))
% Found (x50 (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(L x1):(a->Prop)
% Found x6 as proof of ((L x1) Xy0)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((L Xk) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((L Xk) Xy)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xs:=Xs0:(a->Prop);x1:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x1)))
% Found (x50 x6) as proof of ((L Xy) x1)
% Found ((x5 x1) x6) as proof of ((L Xy) x1)
% Found ((x5 x1) x6) as proof of ((L Xy) x1)
% Found ((x5 x1) x6) as proof of ((L Xy) x1)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xs:=Xs0:(a->Prop);Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x6) as proof of ((L x1) Xy)
% Found ((x5 Xy) x6) as proof of ((L x1) Xy)
% Found ((x5 Xy) x6) as proof of ((L x1) Xy)
% Found ((x5 Xy) x6) as proof of ((L x1) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a
% Found x6 as proof of ((L Xk) Xy0)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x500:=(x50 x4):((L (U Xs)) x1)
% Found (x50 x4) as proof of ((L Xy0) x1)
% Found ((x5 x1) x4) as proof of ((L Xy0) x1)
% Found ((x5 x1) x4) as proof of ((L Xy0) x1)
% Found ((x5 x1) x4) as proof of ((L Xy0) x1)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((L Xk) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a
% Found x6 as proof of ((L Xk) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(L Xk):(a->Prop)
% Found x6 as proof of ((L Xk) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found x500:=(x50 x4):((L (U Xs)) x1)
% Found (x50 x4) as proof of ((L Xy0) x1)
% Found ((x5 x1) x4) as proof of ((L Xy0) x1)
% Found ((x5 x1) x4) as proof of ((L Xy0) x1)
% Found ((x5 x1) x4) as proof of ((L Xy0) x1)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(L Xk):(a->Prop)
% Found x6 as proof of ((L Xk) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xs:=Xs0:(a->Prop);x1:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x1)))
% Found (x50 x6) as proof of ((L Xy) x1)
% Found ((x5 x1) x6) as proof of ((L Xy) x1)
% Found ((x5 x1) x6) as proof of ((L Xy) x1)
% Found ((x5 x1) x6) as proof of ((L Xy) x1)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xs:=Xs0:(a->Prop);Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x6) as proof of ((L x1) Xy)
% Found ((x5 Xy) x6) as proof of ((L x1) Xy)
% Found ((x5 Xy) x6) as proof of ((L x1) Xy)
% Found ((x5 Xy) x6) as proof of ((L x1) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(L x1):(a->Prop)
% Found x6 as proof of ((L x1) Xy0)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x40:=(x4 Xk):((Xs Xk)->((L Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x3))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x3))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of ((Xs Xk)->((L Xk) x3))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x3)))
% Found (x50 (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x3)
% Found ((x5 x3) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x3)
% Found ((x5 x3) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x3)
% Found ((x5 x3) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x3)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: x1:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((L Xk) x1)))
% Found (x70 x6) as proof of ((L Xy) x1)
% Found ((x7 x1) x6) as proof of ((L Xy) x1)
% Found ((x7 x1) x6) as proof of ((L Xy) x1)
% Found ((x7 x1) x6) as proof of ((L Xy) x1)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xs:=Xs0:(a->Prop);Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x6) as proof of ((L x1) Xy)
% Found ((x5 Xy) x6) as proof of ((L x1) Xy)
% Found ((x5 Xy) x6) as proof of ((L x1) Xy)
% Found ((x5 Xy) x6) as proof of ((L x1) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(L x1):(a->Prop)
% Found x6 as proof of ((L x1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found x40:=(x4 Xk):((Xs Xk)->((L Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x3))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x3))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of ((Xs Xk)->((L Xk) x3))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x3)))
% Found (x50 (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x3)
% Found ((x5 x3) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x3)
% Found ((x5 x3) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x3)
% Found ((x5 x3) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x3)
% Found x40:=(x4 Xk):((Xs Xk)->((L Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x1))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x1))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of ((Xs Xk)->((L Xk) x1))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x1)))
% Found (x50 (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((L Xk) Xy)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xs:=Xs0:(a->Prop);x1:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x1)))
% Found (x50 x6) as proof of ((L Xy) x1)
% Found ((x5 x1) x6) as proof of ((L Xy) x1)
% Found ((x5 x1) x6) as proof of ((L Xy) x1)
% Found ((x5 x1) x6) as proof of ((L Xy) x1)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((L Xk) Xy)))
% Found (x70 x6) as proof of ((L x1) Xy)
% Found ((x7 Xy) x6) as proof of ((L x1) Xy)
% Found ((x7 Xy) x6) as proof of ((L x1) Xy)
% Found ((x7 Xy) x6) as proof of ((L x1) Xy)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found x40:=(x4 Xk):((Xs Xk)->((L Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x1))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x1))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of ((Xs Xk)->((L Xk) x1))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x1)))
% Found (x50 (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((L Xk) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a
% Found x6 as proof of ((L Xk) Xy0)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(L Xk):(a->Prop)
% Found x6 as proof of ((L Xk) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found x40:=(x4 Xk):((Xs Xk)->((L Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x1))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x1))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of ((Xs Xk)->((L Xk) x1))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x1)))
% Found (x50 (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found x40:=(x4 Xk):((Xs Xk)->((L Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x1))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x1))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of ((Xs Xk)->((L Xk) x1))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x1)))
% Found (x50 (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((L Xk) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(L x3):(a->Prop)
% Found x6 as proof of ((L x3) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(L x1):(a->Prop)
% Found x6 as proof of ((L x1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found x40:=(x4 Xk):((Xs Xk)->((L Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x3))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x3))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of ((Xs Xk)->((L Xk) x3))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x3)))
% Found (x50 (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x3)
% Found ((x5 x3) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x3)
% Found ((x5 x3) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x3)
% Found ((x5 x3) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x3)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((conj20 ((x5 Xy0) x4)) ((x5 Xy) x4)) as proof of ((and ((L x3) Xy0)) ((L Xy0) Xy))
% Found (((conj2 ((L Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4)) as proof of ((and ((L x3) Xy0)) ((L Xy0) Xy))
% Found ((((conj ((L x3) Xy0)) ((L Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4)) as proof of ((and ((L x3) Xy0)) ((L Xy0) Xy))
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((((conj ((L x3) Xy0)) ((L Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4))) as proof of ((and ((L x3) Xy0)) ((L Xy0) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((((conj ((L x3) Xy0)) ((L Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4))) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((and ((L x3) Xy0)) ((L Xy0) Xy)))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((((conj ((L x3) Xy0)) ((L Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4))) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((and ((L x3) Xy0)) ((L Xy0) Xy))))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((((conj ((L x3) Xy0)) ((L Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4)))) as proof of ((and ((L x3) Xy0)) ((L Xy0) Xy))
% Found ((and_rect1 ((and ((L x3) Xy0)) ((L Xy0) Xy))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((((conj ((L x3) Xy0)) ((L Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4)))) as proof of ((and ((L x3) Xy0)) ((L Xy0) Xy))
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x20)) ((and ((L x3) Xy0)) ((L Xy0) Xy))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((((conj ((L x3) Xy0)) ((L Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4)))) as proof of ((and ((L x3) Xy0)) ((L Xy0) Xy))
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(L x1):(a->Prop)
% Found x6 as proof of ((L x1) Xy0)
% Found x40:=(x4 Xk):((Xs Xk)->((L Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x1))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x1))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of ((Xs Xk)->((L Xk) x1))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x1)))
% Found (x50 (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found x500:=(x50 x4):((L (U Xs)) Xy0)
% Found (x50 x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((conj20 ((x5 Xy0) x4)) ((x5 Xy) x4)) as proof of ((and ((L x1) Xy0)) ((L Xy0) Xy))
% Found (((conj2 ((L Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4)) as proof of ((and ((L x1) Xy0)) ((L Xy0) Xy))
% Found ((((conj ((L x1) Xy0)) ((L Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4)) as proof of ((and ((L x1) Xy0)) ((L Xy0) Xy))
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((((conj ((L x1) Xy0)) ((L Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4))) as proof of ((and ((L x1) Xy0)) ((L Xy0) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((((conj ((L x1) Xy0)) ((L Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4))) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((and ((L x1) Xy0)) ((L Xy0) Xy)))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((((conj ((L x1) Xy0)) ((L Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4))) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((and ((L x1) Xy0)) ((L Xy0) Xy))))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((((conj ((L x1) Xy0)) ((L Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4)))) as proof of ((and ((L x1) Xy0)) ((L Xy0) Xy))
% Found ((and_rect1 ((and ((L x1) Xy0)) ((L Xy0) Xy))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((((conj ((L x1) Xy0)) ((L Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4)))) as proof of ((and ((L x1) Xy0)) ((L Xy0) Xy))
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x30)) ((and ((L x1) Xy0)) ((L Xy0) Xy))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((((conj ((L x1) Xy0)) ((L Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4)))) as proof of ((and ((L x1) Xy0)) ((L Xy0) Xy))
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((L x3) Xy0)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L x3) Xy0))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L x3) Xy0)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((L x3) Xy0)
% Found ((and_rect1 ((L x3) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((L x3) Xy0)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x20)) ((L x3) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((L x3) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: x3:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x3)))
% Found (x50 x4) as proof of ((L Xy0) x3)
% Found ((x5 x3) x4) as proof of ((L Xy0) x3)
% Found ((x5 x3) x4) as proof of ((L Xy0) x3)
% Found ((x5 x3) x4) as proof of ((L Xy0) x3)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((L x1) Xy0)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L x1) Xy0))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L x1) Xy0)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((L x1) Xy0)
% Found ((and_rect1 ((L x1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((L x1) Xy0)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x30)) ((L x1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((L x1) Xy0)
% Found x40:=(x4 Xk):((Xs Xk)->((L Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x3))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x3))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of ((Xs Xk)->((L Xk) x3))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x3)))
% Found (x50 (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x3)
% Found ((x5 x3) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x3)
% Found ((x5 x3) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x3)
% Found ((x5 x3) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x3)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: x1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x1)))
% Found (x50 x4) as proof of ((L Xy0) x1)
% Found ((x5 x1) x4) as proof of ((L Xy0) x1)
% Found ((x5 x1) x4) as proof of ((L Xy0) x1)
% Found ((x5 x1) x4) as proof of ((L Xy0) x1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((L Xy0) Xy)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy0) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy0) Xy)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((L Xy0) Xy)
% Found ((and_rect1 ((L Xy0) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((L Xy0) Xy)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x20)) ((L Xy0) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((L Xy0) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found x40:=(x4 Xk):((Xs Xk)->((L Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x1))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x1))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of ((Xs Xk)->((L Xk) x1))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x1)))
% Found (x50 (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((L Xy0) Xy)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy0) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy0) Xy)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((L Xy0) Xy)
% Found ((and_rect1 ((L Xy0) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((L Xy0) Xy)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x30)) ((L Xy0) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((L Xy0) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((L x3) Xy0)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L x3) Xy0))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L x3) Xy0)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((L x3) Xy0)
% Found ((and_rect1 ((L x3) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((L x3) Xy0)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x20)) ((L x3) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((L x3) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((L Xy0) Xy)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy0) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy0) Xy)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((L Xy0) Xy)
% Found ((and_rect1 ((L Xy0) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((L Xy0) Xy)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x20)) ((L Xy0) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((L Xy0) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((conj20 ((x5 Xy0) x4)) ((x5 Xy) x4)) as proof of ((and ((L x1) Xy0)) ((L Xy0) Xy))
% Found (((conj2 ((L Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4)) as proof of ((and ((L x1) Xy0)) ((L Xy0) Xy))
% Found ((((conj ((L x1) Xy0)) ((L Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4)) as proof of ((and ((L x1) Xy0)) ((L Xy0) Xy))
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((((conj ((L x1) Xy0)) ((L Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4))) as proof of ((and ((L x1) Xy0)) ((L Xy0) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((((conj ((L x1) Xy0)) ((L Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4))) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((and ((L x1) Xy0)) ((L Xy0) Xy)))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((((conj ((L x1) Xy0)) ((L Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4))) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((and ((L x1) Xy0)) ((L Xy0) Xy))))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((((conj ((L x1) Xy0)) ((L Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4)))) as proof of ((and ((L x1) Xy0)) ((L Xy0) Xy))
% Found ((and_rect1 ((and ((L x1) Xy0)) ((L Xy0) Xy))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((((conj ((L x1) Xy0)) ((L Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4)))) as proof of ((and ((L x1) Xy0)) ((L Xy0) Xy))
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x30)) ((and ((L x1) Xy0)) ((L Xy0) Xy))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((((conj ((L x1) Xy0)) ((L Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4)))) as proof of ((and ((L x1) Xy0)) ((L Xy0) Xy))
% Found x40:=(x4 Xk):((Xs Xk)->((L Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x1))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x1))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of ((Xs Xk)->((L Xk) x1))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x1)))
% Found (x50 (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((L x1) Xy0)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L x1) Xy0))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L x1) Xy0)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((L x1) Xy0)
% Found ((and_rect1 ((L x1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((L x1) Xy0)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x30)) ((L x1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((L x1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((L Xy0) Xy)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy0) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy0) Xy)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((L Xy0) Xy)
% Found ((and_rect1 ((L Xy0) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((L Xy0) Xy)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x30)) ((L Xy0) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((L Xy0) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((L x1) Xy0)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L x1) Xy0))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L x1) Xy0)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((L x1) Xy0)
% Found ((and_rect1 ((L x1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((L x1) Xy0)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x30)) ((L x1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((L x1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: x3:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x3)))
% Found (x50 x4) as proof of ((L Xy0) x3)
% Found ((x5 x3) x4) as proof of ((L Xy0) x3)
% Found ((x5 x3) x4) as proof of ((L Xy0) x3)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x3) x4)) as proof of ((L Xy0) x3)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x3) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy0) x3))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x3) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy0) x3)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x3) x4))) as proof of ((L Xy0) x3)
% Found ((and_rect1 ((L Xy0) x3)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x3) x4))) as proof of ((L Xy0) x3)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x20)) ((L Xy0) x3)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x3) x4))) as proof of ((L Xy0) x3)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x3) Xy0)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((L x3) Xy0)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L x3) Xy0))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L x3) Xy0)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((L x3) Xy0)
% Found ((and_rect1 ((L x3) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((L x3) Xy0)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x20)) ((L x3) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((L x3) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: x1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x1)))
% Found (x50 x4) as proof of ((L Xy0) x1)
% Found ((x5 x1) x4) as proof of ((L Xy0) x1)
% Found ((x5 x1) x4) as proof of ((L Xy0) x1)
% Found ((x5 x1) x4) as proof of ((L Xy0) x1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found x40:=(x4 Xk):((Xs Xk)->((L Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x3))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x3))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of ((Xs Xk)->((L Xk) x3))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x3)))
% Found (x50 (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x3)
% Found ((x5 x3) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x3)
% Found ((x5 x3) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x3)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x3) (fun (Xk:a)=> (x4 Xk)))) as proof of ((L Xy0) x3)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x3) (fun (Xk:a)=> (x4 Xk)))) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy0) x3))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x3) (fun (Xk:a)=> (x4 Xk)))) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy0) x3)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x3) (fun (Xk:a)=> (x4 Xk))))) as proof of ((L Xy0) x3)
% Found ((and_rect1 ((L Xy0) x3)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x3) (fun (Xk:a)=> (x4 Xk))))) as proof of ((L Xy0) x3)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x20)) ((L Xy0) x3)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x3) (fun (Xk:a)=> (x4 Xk))))) as proof of ((L Xy0) x3)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: x1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x1)))
% Found (x50 x4) as proof of ((L Xy0) x1)
% Found ((x5 x1) x4) as proof of ((L Xy0) x1)
% Found ((x5 x1) x4) as proof of ((L Xy0) x1)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) x4)) as proof of ((L Xy0) x1)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy0) x1))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy0) x1)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) x4))) as proof of ((L Xy0) x1)
% Found ((and_rect1 ((L Xy0) x1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) x4))) as proof of ((L Xy0) x1)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x30)) ((L Xy0) x1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) x4))) as proof of ((L Xy0) x1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((L x1) Xy0)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L x1) Xy0))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L x1) Xy0)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((L x1) Xy0)
% Found ((and_rect1 ((L x1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((L x1) Xy0)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x30)) ((L x1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((L x1) Xy0)
% Found x40:=(x4 Xk):((Xs Xk)->((L Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x1))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x1))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of ((Xs Xk)->((L Xk) x1))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x1)))
% Found (x50 (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((L Xy0) Xy)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy0) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy0) Xy)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((L Xy0) Xy)
% Found ((and_rect1 ((L Xy0) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((L Xy0) Xy)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x30)) ((L Xy0) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((L Xy0) Xy)
% Found x40:=(x4 Xk):((Xs Xk)->((L Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x1))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x1))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of ((Xs Xk)->((L Xk) x1))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x1)))
% Found (x50 (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) (fun (Xk:a)=> (x4 Xk)))) as proof of ((L Xy0) x1)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) (fun (Xk:a)=> (x4 Xk)))) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy0) x1))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) (fun (Xk:a)=> (x4 Xk)))) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy0) x1)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) (fun (Xk:a)=> (x4 Xk))))) as proof of ((L Xy0) x1)
% Found ((and_rect1 ((L Xy0) x1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) (fun (Xk:a)=> (x4 Xk))))) as proof of ((L Xy0) x1)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x30)) ((L Xy0) x1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) (fun (Xk:a)=> (x4 Xk))))) as proof of ((L Xy0) x1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((L Xy0) Xy)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((L Xy0) Xy)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy0) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy0) Xy)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((L Xy0) Xy)
% Found ((and_rect1 ((L Xy0) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((L Xy0) Xy)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x30)) ((L Xy0) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((L Xy0) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((L x1) Xy0)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L x1) Xy0))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L x1) Xy0)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((L x1) Xy0)
% Found ((and_rect1 ((L x1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((L x1) Xy0)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x30)) ((L x1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((L x1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: x1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x1)))
% Found (x50 x4) as proof of ((L Xy0) x1)
% Found ((x5 x1) x4) as proof of ((L Xy0) x1)
% Found ((x5 x1) x4) as proof of ((L Xy0) x1)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) x4)) as proof of ((L Xy0) x1)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy0) x1))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy0) x1)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) x4))) as proof of ((L Xy0) x1)
% Found ((and_rect1 ((L Xy0) x1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) x4))) as proof of ((L Xy0) x1)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x30)) ((L Xy0) x1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) x4))) as proof of ((L Xy0) x1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x50 x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((L x1) Xy0)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((L x1) Xy0)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L x1) Xy0))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L x1) Xy0)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((L x1) Xy0)
% Found ((and_rect1 ((L x1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((L x1) Xy0)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x30)) ((L x1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((L x1) Xy0)
% Found x40:=(x4 Xk):((Xs Xk)->((L Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x1))
% Found (x4 Xk) as proof of ((Xs Xk)->((L Xk) x1))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of ((Xs Xk)->((L Xk) x1))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x1)))
% Found (x50 (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found ((x5 x1) (fun (Xk:a)=> (x4 Xk))) as proof of ((L Xy0) x1)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) (fun (Xk:a)=> (x4 Xk)))) as proof of ((L Xy0) x1)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) (fun (Xk:a)=> (x4 Xk)))) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy0) x1))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) (fun (Xk:a)=> (x4 Xk)))) as proof of ((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->((L Xy0) x1)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) (fun (Xk:a)=> (x4 Xk))))) as proof of ((L Xy0) x1)
% Found ((and_rect1 ((L Xy0) x1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) (fun (Xk:a)=> (x4 Xk))))) as proof of ((L Xy0) x1)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj)))) P) x4) x30)) ((L Xy0) x1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((L Xk) Xj)))->((L (U Xs)) Xj))))=> ((x5 x1) (fun (Xk:a)=> (x4 Xk))))) as proof of ((L Xy0) x1)
% Found x7:(forall (Xz:a), ((Xs1 Xz)->((L Xz) (U Xs1))))
% Instantiate: Xs:=Xs1:(a->Prop);x9:=(U Xs1):a
% Found x7 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x9)))
% Found (x40 x7) as proof of ((L Xy) x9)
% Found ((x4 x9) x7) as proof of ((L Xy) x9)
% Found ((x4 x9) x7) as proof of ((L Xy) x9)
% Found ((x4 x9) x7) as proof of ((L Xy) x9)
% Found x7:(forall (Xz:a), ((Xs1 Xz)->((L Xz) (U Xs1))))
% Instantiate: Xs0:=Xs1:(a->Prop);Xy:=(U Xs1):a
% Found x7 as proof of (forall (Xk:a), ((Xs0 Xk)->((L Xk) Xy)))
% Found (x60 x7) as proof of ((L x9) Xy)
% Found ((x6 Xy) x7) as proof of ((L x9) Xy)
% Found ((x6 Xy) x7) as proof of ((L x9) Xy)
% Found ((x6 Xy) x7) as proof of ((L x9) Xy)
% Found x10:(Xs1 Xk)
% Instantiate: Xs1:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((L Xk) Xy)
% Found x10:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((L Xk) Xy)
% Found x10:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((L Xk) Xy)
% Found x8:(forall (Xz:a), ((Xs1 Xz)->((L Xz) (U Xs1))))
% Instantiate: Xs0:=Xs1:(a->Prop);x7:=(U Xs1):a
% Found x8 as proof of (forall (Xk:a), ((Xs0 Xk)->((L Xk) x7)))
% Found (x60 x8) as proof of ((L Xy) x7)
% Found ((x6 x7) x8) as proof of ((L Xy) x7)
% Found ((x6 x7) x8) as proof of ((L Xy) x7)
% Found ((x6 x7) x8) as proof of ((L Xy) x7)
% Found x8:(forall (Xz:a), ((Xs1 Xz)->((L Xz) (U Xs1))))
% Instantiate: Xy:=(U Xs1):a
% Found x8 as proof of (forall (Xk:a), ((Xs1 Xk)->((L Xk) Xy)))
% Found (x90 x8) as proof of ((L x7) Xy)
% Found ((x9 Xy) x8) as proof of ((L x7) Xy)
% Found ((x9 Xy) x8) as proof of ((L x7) Xy)
% Found ((x9 Xy) x8) as proof of ((L x7) Xy)
% Found x10:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((L Xk) Xy)
% Found x10:(Xs1 Xk)
% Instantiate: Xs1:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((L Xk) Xy)
% Found x10:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((L Xk) Xy)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xs:=Xs0:(a->Prop);x5:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x5)))
% Found (x40 x6) as proof of ((L Xy) x5)
% Found ((x4 x5) x6) as proof of ((L Xy) x5)
% Found ((x4 x5) x6) as proof of ((L Xy) x5)
% Found ((x4 x5) x6) as proof of ((L Xy) x5)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xs1:=Xs0:(a->Prop);Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs1 Xk)->((L Xk) Xy)))
% Found (x90 x6) as proof of ((L x5) Xy)
% Found ((x9 Xy) x6) as proof of ((L x5) Xy)
% Found ((x9 Xy) x6) as proof of ((L x5) Xy)
% Found ((x9 Xy) x6) as proof of ((L x5) Xy)
% Found x3:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x3 as proof of (forall (Xk:a), ((Xs0 Xk)->((L Xk) Xy0)))
% Found (x60 x3) as proof of ((L x7) Xy0)
% Found ((x6 Xy0) x3) as proof of ((L x7) Xy0)
% Found ((x6 Xy0) x3) as proof of ((L x7) Xy0)
% Found x600:=(x60 x5):((L (U Xs0)) x7)
% Found (x60 x5) as proof of ((L Xy0) x7)
% Found ((x6 x7) x5) as proof of ((L Xy0) x7)
% Found ((x6 x7) x5) as proof of ((L Xy0) x7)
% Found ((x6 x7) x5) as proof of ((L Xy0) x7)
% Found x10:(Xs1 Xk)
% Instantiate: Xs1:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((L Xk) Xy)
% Found x10:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((L Xk) Xy)
% Found x10:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((L Xk) Xy)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xs:=Xs0:(a->Prop);x3:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) x3)))
% Found (x50 x6) as proof of ((L Xy) x3)
% Found ((x5 x3) x6) as proof of ((L Xy) x3)
% Found ((x5 x3) x6) as proof of ((L Xy) x3)
% Found ((x5 x3) x6) as proof of ((L Xy) x3)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xs1:=Xs0:(a->Prop);Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs1 Xk)->((L Xk) Xy)))
% Found (x90 x6) as proof of ((L x3) Xy)
% Found ((x9 Xy) x6) as proof of ((L x3) Xy)
% Found ((x9 Xy) x6) as proof of ((L x3) Xy)
% Found ((x9 Xy) x6) as proof of ((L x3) Xy)
% Found x3:(forall (Xz:a), ((Xs Xz)->((L Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x3 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy0)))
% Found (x40 x3) as proof of ((L x5) Xy0)
% Found ((x4 Xy0) x3) as proof of ((L x5) Xy0)
% Found ((x4 Xy0) x3) as proof of ((L x5) Xy0)
% Found x400:=(x40 x6):((L (U Xs)) x5)
% Found (x40 x6) as proof of ((L Xy0) x5)
% Found ((x4 x5) x6) as proof of ((L Xy0) x5)
% Found ((x4 x5) x6) as proof of ((L Xy0) x5)
% Found ((x4 x5) x6) as proof of ((L Xy0) x5)
% Found x10:(Xs1 Xk)
% Instantiate: Xs1:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((L Xk) Xy)
% Found x10:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((L Xk) Xy)
% Found x10:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((L Xk) Xy)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xs1:=Xs0:(a->Prop);x1:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs1 Xk)->((L Xk) x1)))
% Found (x90 x6) as proof of ((L Xy) x1)
% Found ((x9 x1) x6) as proof of ((L Xy) x1)
% Found ((x9 x1) x6) as proof of ((L Xy) x1)
% Found ((x9 x1) x6) as proof of ((L Xy) x1)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xs:=Xs0:(a->Prop);Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((L Xk) Xy)))
% Found (x50 x6) as proof of ((L x1) Xy)
% Found ((x5 Xy) x6) as proof of ((L x1) Xy)
% Found ((x5 Xy) x6) as proof of ((L x1) Xy)
% Found ((x5 Xy) x6) as proof of ((L x1) Xy)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((L Xk) Xy0)))
% Found (x70 x6) as proof of ((L x3) Xy0)
% Found ((x7 Xy0) x6) as proof of ((L x3) Xy0)
% Found ((x7 Xy0) x6) as proof of ((L x3) Xy0)
% Found x700:=(x70 x6):((L (U Xs0)) x3)
% Found (x70 x6) as proof of ((L Xy0) x3)
% Found ((x7 x3) x6) as proof of ((L Xy0) x3)
% Found ((x7 x3) x6) as proof of ((L Xy0) x3)
% Found ((x7 x3) x6) as proof of ((L Xy0) x3)
% Found x10:(Xs Xk)
% Instantiate: Xs:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((L Xk) Xy)
% Found x10:(Xs0 Xk)
% Instantiate: Xs0:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((L Xk) Xy)
% Found x10:(Xs1 Xk)
% Instantiate: Xs1:=(L Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((L Xk) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xy0:=Xk:a;Xs0:=(L x7):(a->Prop)
% Found x8 as proof of ((L x7) Xy0)
% Found x8:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(L x7):(a->Prop)
% Found x8 as proof of ((L x7) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((L Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((L Xk) Xy0)))
% Found
% EOF
%------------------------------------------------------------------------------