%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV055^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n179.star.cs.uiowa.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory   : 32286.75MB
% OS       : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:40 EDT 2014

% Result   : Timeout 300.03s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV055^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n179.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 07:47:51 CDT 2014
% % CPUTime  : 300.03 
% 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 0x1911bd8>, <kernel.Type object at 0x19119e0>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (R:(a->(a->Prop))) (U:((a->Prop)->a)), (((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xs:(a->Prop)), ((and (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))))->(forall (Xf:(a->a)), ((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R (Xf Xx)) (Xf Xy))))->((ex a) (fun (Xw:a)=> ((R Xw) (Xf Xw)))))))) of role conjecture named cTHM402_pme
% Conjecture to prove = (forall (R:(a->(a->Prop))) (U:((a->Prop)->a)), (((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xs:(a->Prop)), ((and (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))))->(forall (Xf:(a->a)), ((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R (Xf Xx)) (Xf Xy))))->((ex a) (fun (Xw:a)=> ((R Xw) (Xf Xw)))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (R:(a->(a->Prop))) (U:((a->Prop)->a)), (((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xs:(a->Prop)), ((and (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))))->(forall (Xf:(a->a)), ((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R (Xf Xx)) (Xf Xy))))->((ex a) (fun (Xw:a)=> ((R Xw) (Xf Xw))))))))']
% Parameter a:Type.
% Trying to prove (forall (R:(a->(a->Prop))) (U:((a->Prop)->a)), (((and (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) (forall (Xs:(a->Prop)), ((and (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))))->(forall (Xf:(a->a)), ((forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R (Xf Xx)) (Xf Xy))))->((ex a) (fun (Xw:a)=> ((R Xw) (Xf Xw))))))))
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((R Xk) Xy)
% Found x3:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x3 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy)))
% Found (x40 x3) as proof of ((R x5) Xy)
% Found ((x4 Xy) x3) as proof of ((R x5) Xy)
% Found ((x4 Xy) x3) as proof of ((R x5) Xy)
% Found ((x4 Xy) x3) as proof of ((R x5) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((R Xk) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy)))
% Found (x50 x4) as proof of ((R x3) Xy)
% Found ((x5 Xy) x4) as proof of ((R x3) Xy)
% Found ((x5 Xy) x4) as proof of ((R x3) Xy)
% Found ((x5 Xy) x4) as proof of ((R x3) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((R Xk) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy)))
% Found (x50 x4) as proof of ((R x1) Xy)
% Found ((x5 Xy) x4) as proof of ((R x1) Xy)
% Found ((x5 Xy) x4) as proof of ((R x1) Xy)
% Found ((x5 Xy) x4) as proof of ((R x1) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy)))
% Found (x50 x4) as proof of ((R x3) Xy)
% Found ((x5 Xy) x4) as proof of ((R x3) Xy)
% Found ((x5 Xy) x4) as proof of ((R x3) Xy)
% Found ((x5 Xy) x4) as proof of ((R x3) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy)))
% Found (x50 x4) as proof of ((R x1) Xy)
% Found ((x5 Xy) x4) as proof of ((R x1) Xy)
% Found ((x5 Xy) x4) as proof of ((R x1) Xy)
% Found ((x5 Xy) x4) as proof of ((R x1) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy)))
% Found (x50 x4) as proof of ((R x3) Xy)
% Found ((x5 Xy) x4) as proof of ((R x3) Xy)
% Found ((x5 Xy) x4) as proof of ((R x3) Xy)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((R x3) Xy)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((R x3) Xy)
% Found ((and_rect1 ((R x3) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((R x3) Xy)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x20)) ((R x3) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((R x3) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy)))
% Found (x50 x4) as proof of ((R x1) Xy)
% Found ((x5 Xy) x4) as proof of ((R x1) Xy)
% Found ((x5 Xy) x4) as proof of ((R x1) Xy)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((R x1) Xy)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((R x1) Xy)
% Found ((and_rect1 ((R x1) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((R x1) Xy)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x30)) ((R x1) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((R x1) Xy)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((R Xk) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((R Xk) Xy)
% Found x5:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy:=(U Xs0):a
% Found x5 as proof of (forall (Xk:a), ((Xs0 Xk)->((R Xk) Xy)))
% Found (x60 x5) as proof of ((R x7) Xy)
% Found ((x6 Xy) x5) as proof of ((R x7) Xy)
% Found ((x6 Xy) x5) as proof of ((R x7) Xy)
% Found ((x6 Xy) x5) as proof of ((R x7) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x5):(a->Prop)
% Found x6 as proof of ((R x5) Xy0)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((R Xk) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((R Xk) Xy)
% Found x3:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x3 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x40 x3) as proof of ((R x5) Xy0)
% Found ((x4 Xy0) x3) as proof of ((R x5) Xy0)
% Found ((x4 Xy0) x3) as proof of ((R x5) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xs:=Xs0:(a->Prop);Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy)))
% Found (x40 x6) as proof of ((R x5) Xy)
% Found ((x4 Xy) x6) as proof of ((R x5) Xy)
% Found ((x4 Xy) x6) as proof of ((R x5) Xy)
% Found ((x4 Xy) x6) as proof of ((R x5) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x3):(a->Prop)
% Found x6 as proof of ((R x3) Xy0)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((R Xk) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((R Xk) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((R Xk) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x3) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((R Xk) Xy)))
% Found (x70 x6) as proof of ((R x5) Xy)
% Found ((x7 Xy) x6) as proof of ((R x5) Xy)
% Found ((x7 Xy) x6) as proof of ((R x5) Xy)
% Found ((x7 Xy) x6) as proof of ((R x5) Xy)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((R Xk) Xy)))
% Found (x70 x6) as proof of ((R x3) Xy)
% Found ((x7 Xy) x6) as proof of ((R x3) Xy)
% Found ((x7 Xy) x6) as proof of ((R x3) Xy)
% Found ((x7 Xy) x6) as proof of ((R x3) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x1):(a->Prop)
% Found x6 as proof of ((R x1) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x3):(a->Prop)
% Found x6 as proof of ((R x3) Xy0)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((R Xk) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((R Xk) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((R Xk) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((R Xk) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x3) Xy0)
% Found x3:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x3 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x40 x3) as proof of ((R x5) Xy0)
% Found ((x4 Xy0) x3) as proof of ((R x5) Xy0)
% Found ((x4 Xy0) x3) as proof of ((R x5) Xy0)
% Found ((x4 Xy0) x3) as proof of ((R x5) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xs:=Xs0:(a->Prop);Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy)))
% Found (x50 x6) as proof of ((R x3) Xy)
% Found ((x5 Xy) x6) as proof of ((R x3) Xy)
% Found ((x5 Xy) x6) as proof of ((R x3) Xy)
% Found ((x5 Xy) x6) as proof of ((R x3) Xy)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((R Xk) Xy)))
% Found (x70 x6) as proof of ((R x1) Xy)
% Found ((x7 Xy) x6) as proof of ((R x1) Xy)
% Found ((x7 Xy) x6) as proof of ((R x1) Xy)
% Found ((x7 Xy) x6) as proof of ((R x1) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x1):(a->Prop)
% Found x6 as proof of ((R x1) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a
% Found x6 as proof of ((R Xk) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((R Xk) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((R Xk) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x3) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((R Xk) Xy)))
% Found (x70 x6) as proof of ((R x1) Xy)
% Found ((x7 Xy) x6) as proof of ((R x1) Xy)
% Found ((x7 Xy) x6) as proof of ((R x1) Xy)
% Found ((x7 Xy) x6) as proof of ((R x1) Xy)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((R Xk) Xy)))
% Found (x70 x6) as proof of ((R x3) Xy)
% Found ((x7 Xy) x6) as proof of ((R x3) Xy)
% Found ((x7 Xy) x6) as proof of ((R x3) Xy)
% Found ((x7 Xy) x6) as proof of ((R x3) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R Xk):(a->Prop)
% Found x6 as proof of ((R Xk) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a
% Found x6 as proof of ((R Xk) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((R Xk) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x3) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((R Xk) Xy)))
% Found (x70 x6) as proof of ((R x1) Xy)
% Found ((x7 Xy) x6) as proof of ((R x1) Xy)
% Found ((x7 Xy) x6) as proof of ((R x1) Xy)
% Found ((x7 Xy) x6) as proof of ((R x1) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R Xk):(a->Prop)
% Found x6 as proof of ((R Xk) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a
% Found x6 as proof of ((R Xk) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x1) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R Xk):(a->Prop)
% Found x6 as proof of ((R Xk) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x3) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x3):(a->Prop)
% Found x6 as proof of ((R x3) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x1) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x1):(a->Prop)
% Found x6 as proof of ((R x1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x3) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x3) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy)))
% Found (x50 x4) as proof of ((R Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((R Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((R Xy0) Xy)
% Found ((conj10 ((x5 Xy0) x4)) ((x5 Xy) x4)) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found (((conj1 ((R Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4)) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found ((((conj ((R x3) Xy0)) ((R Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4)) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy0)) ((R Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy0)) ((R Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4))) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((and ((R x3) Xy0)) ((R Xy0) Xy)))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy0)) ((R Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4))) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((and ((R x3) Xy0)) ((R Xy0) Xy))))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy0)) ((R Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4)))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found ((and_rect1 ((and ((R x3) Xy0)) ((R Xy0) Xy))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy0)) ((R Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4)))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x20)) ((and ((R x3) Xy0)) ((R Xy0) Xy))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy0)) ((R Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4)))) as proof of ((and ((R x3) Xy0)) ((R Xy0) Xy))
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy)))
% Found (x50 x4) as proof of ((R Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((R Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((R Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((R Xy0) Xy)
% Found x500:=(x50 x4):((R (U Xs)) Xy0)
% Found (x50 x4) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x1) Xy0)
% Found ((conj10 ((x5 Xy0) x4)) ((x5 Xy) x4)) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found (((conj1 ((R Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4)) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((((conj ((R x1) Xy0)) ((R Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4)) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy0)) ((R Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy0)) ((R Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4))) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((and ((R x1) Xy0)) ((R Xy0) Xy)))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy0)) ((R Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4))) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((and ((R x1) Xy0)) ((R Xy0) Xy))))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy0)) ((R Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4)))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found ((and_rect1 ((and ((R x1) Xy0)) ((R Xy0) Xy))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy0)) ((R Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4)))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x30)) ((and ((R x1) Xy0)) ((R Xy0) Xy))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy0)) ((R Xy0) Xy)) ((x5 Xy0) x4)) ((x5 Xy) x4)))) as proof of ((and ((R x1) Xy0)) ((R Xy0) Xy))
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x3) Xy0)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((R x3) Xy0)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy0))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy0)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R x3) Xy0)
% Found ((and_rect1 ((R x3) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R x3) Xy0)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x20)) ((R x3) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R x3) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x3) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x1) Xy0)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((R x1) Xy0)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy0))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy0)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R x1) Xy0)
% Found ((and_rect1 ((R x1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R x1) Xy0)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x30)) ((R x1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R x1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x3) Xy0)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((R x3) Xy0)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy0))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy0)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R x3) Xy0)
% Found ((and_rect1 ((R x3) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R x3) Xy0)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x20)) ((R x3) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R x3) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy)))
% Found (x50 x4) as proof of ((R Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((R Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((R Xy0) Xy)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((R Xy0) Xy)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R Xy0) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R Xy0) Xy)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((R Xy0) Xy)
% Found ((and_rect1 ((R Xy0) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((R Xy0) Xy)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x20)) ((R Xy0) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((R Xy0) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy)))
% Found (x50 x4) as proof of ((R Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((R Xy0) Xy)
% Found ((x5 Xy) x4) as proof of ((R Xy0) Xy)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((R Xy0) Xy)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R Xy0) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R Xy0) Xy)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((R Xy0) Xy)
% Found ((and_rect1 ((R Xy0) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((R Xy0) Xy)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x30)) ((R Xy0) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((R Xy0) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x1) Xy0)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((R x1) Xy0)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy0))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy0)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R x1) Xy0)
% Found ((and_rect1 ((R x1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R x1) Xy0)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x30)) ((R x1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R x1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x3) Xy0)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((R x3) Xy0)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy0))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy0)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R x3) Xy0)
% Found ((and_rect1 ((R x3) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R x3) Xy0)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x20)) ((R x3) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R x3) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R x1) Xy0)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((R x1) Xy0)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy0))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy0)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R x1) Xy0)
% Found ((and_rect1 ((R x1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R x1) Xy0)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x30)) ((R x1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R x1) Xy0)
% Found x8:(Xs0 Xk)
% Instantiate: Xy0:=Xk:a;Xs0:=(R x7):(a->Prop)
% Found x8 as proof of ((R x7) Xy0)
% Found x8:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x7):(a->Prop)
% Found x8 as proof of ((R x7) Xy0)
% Found x3:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a;Xs0:=Xs:(a->Prop)
% Found x3 as proof of (forall (Xk:a), ((Xs0 Xk)->((R Xk) Xy0)))
% Found (x60 x3) as proof of ((R x7) Xy0)
% Found ((x6 Xy0) x3) as proof of ((R x7) Xy0)
% Found ((x6 Xy0) x3) as proof of ((R x7) Xy0)
% Found x10:(Xs1 Xk)
% Instantiate: Xs1:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((R Xk) Xy)
% Found x10:(Xs0 Xk)
% Instantiate: Xs0:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((R Xk) Xy)
% Found x10:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((R Xk) Xy)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a;Xs:=Xs0:(a->Prop)
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x40 x6) as proof of ((R x5) Xy0)
% Found ((x4 Xy0) x6) as proof of ((R x5) Xy0)
% Found ((x4 Xy0) x6) as proof of ((R x5) Xy0)
% Found x3:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xs1:=Xs:(a->Prop);Xy:=(U Xs):a
% Found x3 as proof of (forall (Xk:a), ((Xs1 Xk)->((R Xk) Xy)))
% Found (x80 x3) as proof of ((R x9) Xy)
% Found ((x8 Xy) x3) as proof of ((R x9) Xy)
% Found ((x8 Xy) x3) as proof of ((R x9) Xy)
% Found ((x8 Xy) x3) as proof of ((R x9) Xy)
% Found x8:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x5):(a->Prop)
% Found x8 as proof of ((R x5) Xy0)
% Found x8:(Xs0 Xk)
% Instantiate: Xy0:=Xk:a;Xs0:=(R x5):(a->Prop)
% Found x8 as proof of ((R x5) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((R Xk) Xy0)))
% Found (x70 x6) as proof of ((R x5) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x5) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x5) Xy0)
% Found x10:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((R Xk) Xy)
% Found x10:(Xs0 Xk)
% Instantiate: Xs0:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((R Xk) Xy)
% Found x10:(Xs1 Xk)
% Instantiate: Xs1:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((R Xk) Xy)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a;Xs:=Xs0:(a->Prop)
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x6) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x3) Xy0)
% Found x3:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x3 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy)))
% Found (x40 x3) as proof of ((R x7) Xy)
% Found ((x4 Xy) x3) as proof of ((R x7) Xy)
% Found ((x4 Xy) x3) as proof of ((R x7) Xy)
% Found ((x4 Xy) x3) as proof of ((R x7) Xy)
% Found x8:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x5):(a->Prop)
% Found x8 as proof of ((R x5) Xy0)
% Found x8:(Xs0 Xk)
% Instantiate: Xy0:=Xk:a;Xs0:=(R x5):(a->Prop)
% Found x8 as proof of ((R x5) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((R Xk) Xy0)))
% Found (x70 x6) as proof of ((R x5) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x5) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x5) Xy0)
% Found x8:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x3):(a->Prop)
% Found x8 as proof of ((R x3) Xy0)
% Found x8:(Xs0 Xk)
% Instantiate: Xy0:=Xk:a;Xs0:=(R x3):(a->Prop)
% Found x8 as proof of ((R x3) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((R Xk) Xy0)))
% Found (x70 x6) as proof of ((R x3) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x3) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x3) Xy0)
% Found x3:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x3 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x40 x3) as proof of ((R x5) Xy1)
% Found ((x4 Xy1) x3) as proof of ((R x5) Xy1)
% Found ((x4 Xy1) x3) as proof of ((R x5) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xy1:=Xk:a
% Found x6 as proof of ((R x5) Xy1)
% Found x3:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x3 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x40 x3) as proof of ((R x7) Xy0)
% Found ((x4 Xy0) x3) as proof of ((R x7) Xy0)
% Found ((x4 Xy0) x3) as proof of ((R x7) Xy0)
% Found ((x4 Xy0) x3) as proof of ((R x7) Xy0)
% Found x10:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((R Xk) Xy)
% Found x10:(Xs0 Xk)
% Instantiate: Xs0:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((R Xk) Xy)
% Found x10:(Xs1 Xk)
% Instantiate: Xs1:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((R Xk) Xy)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((R Xk) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((R Xk) Xy)
% Found x3:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x3 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x40 x3) as proof of ((R x5) Xy1)
% Found ((x4 Xy1) x3) as proof of ((R x5) Xy1)
% Found ((x4 Xy1) x3) as proof of ((R x5) Xy1)
% Found x3:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x3 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x40 x3) as proof of ((R x5) Xy1)
% Found ((x4 Xy1) x3) as proof of ((R x5) Xy1)
% Found ((x4 Xy1) x3) as proof of ((R x5) Xy1)
% Found x8:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x5):(a->Prop)
% Found x8 as proof of ((R x5) Xy0)
% Found x8:(Xs0 Xk)
% Instantiate: Xy0:=Xk:a;Xs0:=(R x5):(a->Prop)
% Found x8 as proof of ((R x5) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R x5):(a->Prop);Xy1:=Xk:a
% Found x6 as proof of ((R x5) Xy1)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a;Xs:=Xs0:(a->Prop)
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a;Xs:=Xs0:(a->Prop)
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x6) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x3) Xy0)
% Found x3:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xs1:=Xs:(a->Prop);Xy:=(U Xs):a
% Found x3 as proof of (forall (Xk:a), ((Xs1 Xk)->((R Xk) Xy)))
% Found (x90 x3) as proof of ((R x7) Xy)
% Found ((x9 Xy) x3) as proof of ((R x7) Xy)
% Found ((x9 Xy) x3) as proof of ((R x7) Xy)
% Found ((x9 Xy) x3) as proof of ((R x7) Xy)
% Found x8:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x3):(a->Prop)
% Found x8 as proof of ((R x3) Xy0)
% Found x3:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xs1:=Xs:(a->Prop);Xy:=(U Xs):a
% Found x3 as proof of (forall (Xk:a), ((Xs1 Xk)->((R Xk) Xy)))
% Found (x90 x3) as proof of ((R x5) Xy)
% Found ((x9 Xy) x3) as proof of ((R x5) Xy)
% Found ((x9 Xy) x3) as proof of ((R x5) Xy)
% Found ((x9 Xy) x3) as proof of ((R x5) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xy0:=Xk:a;Xs0:=(R x3):(a->Prop)
% Found x8 as proof of ((R x3) Xy0)
% Found x8:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x1):(a->Prop)
% Found x8 as proof of ((R x1) Xy0)
% Found x8:(Xs0 Xk)
% Instantiate: Xy0:=Xk:a;Xs0:=(R x1):(a->Prop)
% Found x8 as proof of ((R x1) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((R Xk) Xy0)))
% Found (x70 x6) as proof of ((R x3) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x3) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x3) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((R Xk) Xy0)))
% Found (x70 x6) as proof of ((R x1) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x1) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xy1:=Xk:a
% Found x6 as proof of ((R x3) Xy1)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a;Xs:=Xs0:(a->Prop)
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x40 x6) as proof of ((R x5) Xy0)
% Found ((x4 Xy0) x6) as proof of ((R x5) Xy0)
% Found ((x4 Xy0) x6) as proof of ((R x5) Xy0)
% Found ((x4 Xy0) x6) as proof of ((R x5) Xy0)
% Found x8:(Xs Xk)
% Instantiate: Xy0:=Xk:a
% Found x8 as proof of ((R Xk) Xy0)
% Found x8:(Xs0 Xk)
% Instantiate: Xy0:=Xk:a
% Found x8 as proof of ((R Xk) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a;Xs:=Xs0:(a->Prop)
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x40 x6) as proof of ((R x5) Xy0)
% Found ((x4 Xy0) x6) as proof of ((R x5) Xy0)
% Found ((x4 Xy0) x6) as proof of ((R x5) Xy0)
% Found ((x4 Xy0) x6) as proof of ((R x5) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a;Xs:=Xs0:(a->Prop)
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x40 x6) as proof of ((R x5) Xy0)
% Found ((x4 Xy0) x6) as proof of ((R x5) Xy0)
% Found ((x4 Xy0) x6) as proof of ((R x5) Xy0)
% Found x10:(Xs0 Xk)
% Instantiate: Xs0:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((R Xk) Xy)
% Found x10:(Xs1 Xk)
% Instantiate: Xs1:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((R Xk) Xy)
% Found x10:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((R Xk) Xy)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((R Xk) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((R Xk) Xy)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((R Xk) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((R Xk) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found x8:(Xs0 Xk)
% Instantiate: Xy0:=Xk:a;Xs0:=(R x3):(a->Prop)
% Found x8 as proof of ((R x3) Xy0)
% Found x8:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x3):(a->Prop)
% Found x8 as proof of ((R x3) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R x3):(a->Prop);Xy1:=Xk:a
% Found x6 as proof of ((R x3) Xy1)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a;Xs:=Xs0:(a->Prop)
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found x8:(Xs0 Xk)
% Instantiate: Xy0:=Xk:a;Xs0:=(R Xk):(a->Prop)
% Found x8 as proof of ((R Xk) Xy0)
% Found x8:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R Xk):(a->Prop)
% Found x8 as proof of ((R Xk) Xy0)
% Found x3:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xs1:=Xs:(a->Prop);Xy:=(U Xs):a
% Found x3 as proof of (forall (Xk:a), ((Xs1 Xk)->((R Xk) Xy)))
% Found (x90 x3) as proof of ((R x5) Xy)
% Found ((x9 Xy) x3) as proof of ((R x5) Xy)
% Found ((x9 Xy) x3) as proof of ((R x5) Xy)
% Found ((x9 Xy) x3) as proof of ((R x5) Xy)
% Found x8:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x3):(a->Prop)
% Found x8 as proof of ((R x3) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x5):(a->Prop)
% Found x6 as proof of ((R x5) Xy0)
% Found x8:(Xs0 Xk)
% Instantiate: Xy0:=Xk:a;Xs0:=(R x3):(a->Prop)
% Found x8 as proof of ((R x3) Xy0)
% Found x8:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x1):(a->Prop)
% Found x8 as proof of ((R x1) Xy0)
% Found x8:(Xs0 Xk)
% Instantiate: Xy0:=Xk:a;Xs0:=(R x1):(a->Prop)
% Found x8 as proof of ((R x1) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((R Xk) Xy)))
% Found (x70 x6) as proof of ((R x3) Xy)
% Found ((x7 Xy) x6) as proof of ((R x3) Xy)
% Found ((x7 Xy) x6) as proof of ((R x3) Xy)
% Found ((x7 Xy) x6) as proof of ((R x3) Xy)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((R Xk) Xy0)))
% Found (x70 x6) as proof of ((R x3) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x3) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x3) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((R Xk) Xy0)))
% Found (x70 x6) as proof of ((R x1) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x1) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xy1:=Xk:a
% Found x6 as proof of ((R x1) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xy1:=Xk:a
% Found x6 as proof of ((R x3) Xy1)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((R Xk) Xy0)))
% Found (x70 x6) as proof of ((R x3) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x3) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x3) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x3) Xy0)
% Found x8:(Xs0 Xk)
% Instantiate: Xy0:=Xk:a
% Found x8 as proof of ((R Xk) Xy0)
% Found x8:(Xs Xk)
% Instantiate: Xy0:=Xk:a
% Found x8 as proof of ((R Xk) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a;Xs:=Xs0:(a->Prop)
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x40 x6) as proof of ((R x5) Xy0)
% Found ((x4 Xy0) x6) as proof of ((R x5) Xy0)
% Found ((x4 Xy0) x6) as proof of ((R x5) Xy0)
% Found ((x4 Xy0) x6) as proof of ((R x5) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a;Xs:=Xs0:(a->Prop)
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x6) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x3) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a;Xs:=Xs0:(a->Prop)
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x6) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x3) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a
% Found x6 as proof of ((R Xk) Xy0)
% Found x10:(Xs0 Xk)
% Instantiate: Xs0:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((R Xk) Xy)
% Found x10:(Xs1 Xk)
% Instantiate: Xs1:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((R Xk) Xy)
% Found x10:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x10 as proof of ((R Xk) Xy)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((R Xk) Xy)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((R Xk) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((R Xk) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((R Xk) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x5):(a->Prop)
% Found x6 as proof of ((R x5) Xy0)
% Found x8:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x1):(a->Prop)
% Found x8 as proof of ((R x1) Xy0)
% Found x8:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x3):(a->Prop)
% Found x8 as proof of ((R x3) Xy0)
% Found x8:(Xs0 Xk)
% Instantiate: Xy0:=Xk:a;Xs0:=(R x3):(a->Prop)
% Found x8 as proof of ((R x3) Xy0)
% Found x8:(Xs0 Xk)
% Instantiate: Xy0:=Xk:a;Xs0:=(R x1):(a->Prop)
% Found x8 as proof of ((R x1) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R x1):(a->Prop);Xy1:=Xk:a
% Found x6 as proof of ((R x1) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R x3):(a->Prop);Xy1:=Xk:a
% Found x6 as proof of ((R x3) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((R Xk) Xy)
% Found x3:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xs0:=Xs:(a->Prop);Xy:=(U Xs):a
% Found x3 as proof of (forall (Xk:a), ((Xs0 Xk)->((R Xk) Xy)))
% Found (x70 x3) as proof of ((R x5) Xy)
% Found ((x7 Xy) x3) as proof of ((R x5) Xy)
% Found ((x7 Xy) x3) as proof of ((R x5) Xy)
% Found ((x7 Xy) x3) as proof of ((R x5) Xy)
% Found x8:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R Xk):(a->Prop)
% Found x8 as proof of ((R Xk) Xy0)
% Found x8:(Xs0 Xk)
% Instantiate: Xy0:=Xk:a;Xs0:=(R Xk):(a->Prop)
% Found x8 as proof of ((R Xk) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xs:=Xs0:(a->Prop);Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy)))
% Found (x50 x6) as proof of ((R x3) Xy)
% Found ((x5 Xy) x6) as proof of ((R x3) Xy)
% Found ((x5 Xy) x6) as proof of ((R x3) Xy)
% Found ((x5 Xy) x6) as proof of ((R x3) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xy0:=Xk:a;Xs0:=(R x1):(a->Prop)
% Found x8 as proof of ((R x1) Xy0)
% Found x8:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x1):(a->Prop)
% Found x8 as proof of ((R x1) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xs1:=Xs0:(a->Prop);Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs1 Xk)->((R Xk) Xy)))
% Found (x90 x6) as proof of ((R x1) Xy)
% Found ((x9 Xy) x6) as proof of ((R x1) Xy)
% Found ((x9 Xy) x6) as proof of ((R x1) Xy)
% Found ((x9 Xy) x6) as proof of ((R x1) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x3):(a->Prop)
% Found x6 as proof of ((R x3) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((R Xk) Xy0)))
% Found (x70 x6) as proof of ((R x1) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x1) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x1) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R Xk):(a->Prop)
% Found x6 as proof of ((R Xk) Xy0)
% Found x3:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x3 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x40 x3) as proof of ((R x5) Xy1)
% Found ((x4 Xy1) x3) as proof of ((R x5) Xy1)
% Found ((x4 Xy1) x3) as proof of ((R x5) Xy1)
% Found ((x4 Xy1) x3) as proof of ((R x5) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xy1:=Xk:a
% Found x6 as proof of ((R x1) Xy1)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((R Xk) Xy0)))
% Found (x70 x6) as proof of ((R x3) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x3) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x3) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x3) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a;Xs:=Xs0:(a->Prop)
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found x8:(Xs0 Xk)
% Instantiate: Xy0:=Xk:a
% Found x8 as proof of ((R Xk) Xy0)
% Found x8:(Xs Xk)
% Instantiate: Xy0:=Xk:a
% Found x8 as proof of ((R Xk) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a;Xs:=Xs0:(a->Prop)
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x6) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x3) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a;Xs:=Xs0:(a->Prop)
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a;Xs:=Xs0:(a->Prop)
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x6) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x3) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a;Xs:=Xs0:(a->Prop)
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a
% Found x6 as proof of ((R Xk) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a
% Found x6 as proof of ((R Xk) Xy0)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((R Xk) Xy)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((R Xk) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((R Xk) Xy)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((R Xk) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found x8:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x1):(a->Prop)
% Found x8 as proof of ((R x1) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x3):(a->Prop)
% Found x6 as proof of ((R x3) Xy0)
% Found x8:(Xs0 Xk)
% Instantiate: Xy0:=Xk:a;Xs0:=(R x1):(a->Prop)
% Found x8 as proof of ((R x1) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R x1):(a->Prop);Xy1:=Xk:a
% Found x6 as proof of ((R x1) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((R Xk) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((R Xk) Xy)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a;Xs:=Xs0:(a->Prop)
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x40 x6) as proof of ((R x5) Xy0)
% Found ((x4 Xy0) x6) as proof of ((R x5) Xy0)
% Found ((x4 Xy0) x6) as proof of ((R x5) Xy0)
% Found ((x4 Xy0) x6) as proof of ((R x5) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xs1:=Xs0:(a->Prop);Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs1 Xk)->((R Xk) Xy)))
% Found (x90 x6) as proof of ((R x3) Xy)
% Found ((x9 Xy) x6) as proof of ((R x3) Xy)
% Found ((x9 Xy) x6) as proof of ((R x3) Xy)
% Found ((x9 Xy) x6) as proof of ((R x3) Xy)
% Found x8:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R Xk):(a->Prop)
% Found x8 as proof of ((R Xk) Xy0)
% Found x8:(Xs0 Xk)
% Instantiate: Xy0:=Xk:a;Xs0:=(R Xk):(a->Prop)
% Found x8 as proof of ((R Xk) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xs1:=Xs0:(a->Prop);Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs1 Xk)->((R Xk) Xy)))
% Found (x90 x6) as proof of ((R x1) Xy)
% Found ((x9 Xy) x6) as proof of ((R x1) Xy)
% Found ((x9 Xy) x6) as proof of ((R x1) Xy)
% Found ((x9 Xy) x6) as proof of ((R x1) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x1):(a->Prop)
% Found x6 as proof of ((R x1) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x3):(a->Prop)
% Found x6 as proof of ((R x3) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R Xk):(a->Prop)
% Found x6 as proof of ((R Xk) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R Xk):(a->Prop)
% Found x6 as proof of ((R Xk) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a;Xs:=Xs0:(a->Prop)
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found x8:(Xs0 Xk)
% Instantiate: Xy0:=Xk:a
% Found x8 as proof of ((R Xk) Xy0)
% Found x8:(Xs Xk)
% Instantiate: Xy0:=Xk:a
% Found x8 as proof of ((R Xk) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((R Xk) Xy0)))
% Found (x70 x6) as proof of ((R x3) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x3) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x3) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x3) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a;Xs:=Xs0:(a->Prop)
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a;Xs:=Xs0:(a->Prop)
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a
% Found x6 as proof of ((R Xk) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a
% Found x6 as proof of ((R Xk) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a
% Found x6 as proof of ((R Xk) Xy0)
% Found x8:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((R Xk) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xs0:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x8 as proof of ((R Xk) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x1):(a->Prop)
% Found x6 as proof of ((R x1) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x3):(a->Prop)
% Found x6 as proof of ((R x3) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((R Xk) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((R Xk) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((R Xk) Xy)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((R Xk) Xy)))
% Found (x70 x6) as proof of ((R x3) Xy)
% Found ((x7 Xy) x6) as proof of ((R x3) Xy)
% Found ((x7 Xy) x6) as proof of ((R x3) Xy)
% Found ((x7 Xy) x6) as proof of ((R x3) Xy)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a;Xs:=Xs0:(a->Prop)
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x6) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x3) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xs:=Xs0:(a->Prop);Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy)))
% Found (x50 x6) as proof of ((R x1) Xy)
% Found ((x5 Xy) x6) as proof of ((R x1) Xy)
% Found ((x5 Xy) x6) as proof of ((R x1) Xy)
% Found ((x5 Xy) x6) as proof of ((R x1) Xy)
% Found x8:(Xs0 Xk)
% Instantiate: Xy0:=Xk:a;Xs0:=(R Xk):(a->Prop)
% Found x8 as proof of ((R Xk) Xy0)
% Found x8:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R Xk):(a->Prop)
% Found x8 as proof of ((R Xk) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x1):(a->Prop)
% Found x6 as proof of ((R x1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R Xk):(a->Prop)
% Found x6 as proof of ((R Xk) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R Xk):(a->Prop)
% Found x6 as proof of ((R Xk) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R Xk):(a->Prop)
% Found x6 as proof of ((R Xk) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a;Xs:=Xs0:(a->Prop)
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a
% Found x6 as proof of ((R Xk) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a
% Found x6 as proof of ((R Xk) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy1:=Xk:a
% Found x6 as proof of ((R Xk) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x1):(a->Prop)
% Found x6 as proof of ((R x1) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((R Xk) Xy)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xs:=Xs0:(a->Prop);Xy:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy)))
% Found (x50 x6) as proof of ((R x1) Xy)
% Found ((x5 Xy) x6) as proof of ((R x1) Xy)
% Found ((x5 Xy) x6) as proof of ((R x1) Xy)
% Found ((x5 Xy) x6) as proof of ((R x1) Xy)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((R Xk) Xy0)))
% Found (x70 x6) as proof of ((R x1) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x1) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x1) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x1) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a
% Found x6 as proof of ((R Xk) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((R Xk) Xy0)))
% Found (x70 x6) as proof of ((R x3) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x3) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x3) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x3) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((R Xk) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R Xk):(a->Prop)
% Found x6 as proof of ((R Xk) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy1:=Xk:a
% Found x6 as proof of ((R Xk) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R Xk):(a->Prop)
% Found x6 as proof of ((R Xk) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy1:=Xk:a
% Found x6 as proof of ((R Xk) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a
% Found x6 as proof of ((R Xk) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy1:=Xk:a
% Found x6 as proof of ((R Xk) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R Xk):(a->Prop)
% Found x6 as proof of ((R Xk) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy1:=Xk:a
% Found x6 as proof of ((R Xk) Xy1)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((R Xk) Xy0)))
% Found (x70 x6) as proof of ((R x1) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x1) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x1) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x1) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy:=Xk:a
% Found x6 as proof of ((R Xk) Xy)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a
% Found x6 as proof of ((R Xk) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy1:=Xk:a
% Found x6 as proof of ((R Xk) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R Xk):(a->Prop)
% Found x6 as proof of ((R Xk) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy1:=Xk:a
% Found x6 as proof of ((R Xk) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xy1:=Xk:a
% Found x6 as proof of ((R Xk) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xy1:=Xk:a
% Found x6 as proof of ((R Xk) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xy1:=Xk:a
% Found x6 as proof of ((R Xk) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R Xk):(a->Prop)
% Found x6 as proof of ((R Xk) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy1:=Xk:a
% Found x6 as proof of ((R Xk) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a
% Found x6 as proof of ((R Xk) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy1:=Xk:a
% Found x6 as proof of ((R Xk) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xy1:=Xk:a
% Found x6 as proof of ((R Xk) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xy1:=Xk:a
% Found x6 as proof of ((R Xk) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xy1:=Xk:a
% Found x6 as proof of ((R Xk) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R Xk):(a->Prop)
% Found x6 as proof of ((R Xk) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R Xk):(a->Prop);Xy1:=Xk:a
% Found x6 as proof of ((R Xk) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xy1:=Xk:a
% Found x6 as proof of ((R Xk) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xy1:=Xk:a
% Found x6 as proof of ((R Xk) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a;Xs:=Xs0:(a->Prop)
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x6) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x3) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy1:=Xk:a
% Found x6 as proof of ((R x3) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((R Xk) Xy0)))
% Found (x70 x6) as proof of ((R x3) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x3) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x3) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x3) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found x8:(Xs0 Xk)
% Instantiate: Xy0:=Xk:a;Xs0:=(R x3):(a->Prop)
% Found x8 as proof of ((R x3) Xy0)
% Found x8:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x3):(a->Prop)
% Found x8 as proof of ((R x3) Xy0)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a;Xs:=Xs0:(a->Prop)
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy1:=Xk:a
% Found x6 as proof of ((R x1) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R x3):(a->Prop);Xy1:=Xk:a
% Found x6 as proof of ((R x3) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a;Xs:=Xs0:(a->Prop)
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found x8:(Xs0 Xk)
% Instantiate: Xy0:=Xk:a;Xs0:=(R x1):(a->Prop)
% Found x8 as proof of ((R x1) Xy0)
% Found x8:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x1):(a->Prop)
% Found x8 as proof of ((R x1) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x3):(a->Prop)
% Found x6 as proof of ((R x3) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R x1):(a->Prop);Xy1:=Xk:a
% Found x6 as proof of ((R x1) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x3):(a->Prop)
% Found x6 as proof of ((R x3) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x1):(a->Prop)
% Found x6 as proof of ((R x1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a
% Found x6 as proof of (forall (Xk:a), ((Xs0 Xk)->((R Xk) Xy0)))
% Found (x70 x6) as proof of ((R x3) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x3) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x3) Xy0)
% Found ((x7 Xy0) x6) as proof of ((R x3) Xy0)
% Found x6:(Xs Xk)
% Instantiate: Xy0:=Xk:a;Xs:=(R x1):(a->Prop)
% Found x6 as proof of ((R x1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a;Xs:=Xs0:(a->Prop)
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R x3):(a->Prop);Xy1:=Xk:a
% Found x6 as proof of ((R x3) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R x1):(a->Prop);Xy1:=Xk:a
% Found x6 as proof of ((R x1) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a;Xs:=Xs0:(a->Prop)
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x6) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x3) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x3) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found x6:(forall (Xz:a), ((Xs0 Xz)->((R Xz) (U Xs0))))
% Instantiate: Xy0:=(U Xs0):a;Xs:=Xs0:(a->Prop)
% Found x6 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found ((x5 Xy0) x6) as proof of ((R x1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found x500:=(x50 x4):((R (U Xs)) Xy)
% Found (x50 x4) as proof of ((R x3) Xy)
% Found ((x5 Xy) x4) as proof of ((R x3) Xy)
% Found ((x5 Xy) x4) as proof of ((R x3) Xy)
% Found ((x5 Xy) x4) as proof of ((R x3) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found ((conj01 ((x5 Xy) x4)) ((x5 Xy0) x4)) as proof of ((and ((R x3) Xy1)) ((R Xy1) Xy0))
% Found (((conj0 ((R Xy1) Xy0)) ((x5 Xy) x4)) ((x5 Xy0) x4)) as proof of ((and ((R x3) Xy1)) ((R Xy1) Xy0))
% Found (((conj0 ((R Xy1) Xy0)) ((x5 Xy) x4)) ((x5 Xy0) x4)) as proof of ((and ((R x3) Xy1)) ((R Xy1) Xy0))
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> (((conj0 ((R Xy1) Xy0)) ((x5 Xy) x4)) ((x5 Xy0) x4))) as proof of ((and ((R x3) Xy1)) ((R Xy1) Xy0))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> (((conj0 ((R Xy1) Xy0)) ((x5 Xy) x4)) ((x5 Xy0) x4))) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((and ((R x3) Xy1)) ((R Xy1) Xy0)))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> (((conj0 ((R Xy1) Xy0)) ((x5 Xy) x4)) ((x5 Xy0) x4))) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((and ((R x3) Xy1)) ((R Xy1) Xy0))))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> (((conj0 ((R Xy1) Xy0)) ((x5 Xy) x4)) ((x5 Xy0) x4)))) as proof of ((and ((R x3) Xy1)) ((R Xy1) Xy0))
% Found ((and_rect1 ((and ((R x3) Xy1)) ((R Xy1) Xy0))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> (((conj0 ((R Xy1) Xy0)) ((x5 Xy) x4)) ((x5 Xy0) x4)))) as proof of ((and ((R x3) Xy1)) ((R Xy1) Xy0))
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x20)) ((and ((R x3) Xy1)) ((R Xy1) Xy0))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> (((conj0 ((R Xy1) Xy0)) ((x5 Xy) x4)) ((x5 Xy0) x4)))) as proof of ((and ((R x3) Xy1)) ((R Xy1) Xy0))
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found x500:=(x50 x4):((R (U Xs)) Xy)
% Found (x50 x4) as proof of ((R x1) Xy)
% Found ((x5 Xy) x4) as proof of ((R x1) Xy)
% Found ((x5 Xy) x4) as proof of ((R x1) Xy)
% Found ((x5 Xy) x4) as proof of ((R x1) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found ((conj01 ((x5 Xy) x4)) ((x5 Xy0) x4)) as proof of ((and ((R x1) Xy1)) ((R Xy1) Xy0))
% Found (((conj0 ((R Xy1) Xy0)) ((x5 Xy) x4)) ((x5 Xy0) x4)) as proof of ((and ((R x1) Xy1)) ((R Xy1) Xy0))
% Found (((conj0 ((R Xy1) Xy0)) ((x5 Xy) x4)) ((x5 Xy0) x4)) as proof of ((and ((R x1) Xy1)) ((R Xy1) Xy0))
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> (((conj0 ((R Xy1) Xy0)) ((x5 Xy) x4)) ((x5 Xy0) x4))) as proof of ((and ((R x1) Xy1)) ((R Xy1) Xy0))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> (((conj0 ((R Xy1) Xy0)) ((x5 Xy) x4)) ((x5 Xy0) x4))) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((and ((R x1) Xy1)) ((R Xy1) Xy0)))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> (((conj0 ((R Xy1) Xy0)) ((x5 Xy) x4)) ((x5 Xy0) x4))) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((and ((R x1) Xy1)) ((R Xy1) Xy0))))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> (((conj0 ((R Xy1) Xy0)) ((x5 Xy) x4)) ((x5 Xy0) x4)))) as proof of ((and ((R x1) Xy1)) ((R Xy1) Xy0))
% Found ((and_rect1 ((and ((R x1) Xy1)) ((R Xy1) Xy0))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> (((conj0 ((R Xy1) Xy0)) ((x5 Xy) x4)) ((x5 Xy0) x4)))) as proof of ((and ((R x1) Xy1)) ((R Xy1) Xy0))
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x30)) ((and ((R x1) Xy1)) ((R Xy1) Xy0))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> (((conj0 ((R Xy1) Xy0)) ((x5 Xy) x4)) ((x5 Xy0) x4)))) as proof of ((and ((R x1) Xy1)) ((R Xy1) Xy0))
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R x3):(a->Prop);Xy1:=Xk:a
% Found x6 as proof of ((R x3) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((R x3) Xy1)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy1))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy1)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x3) Xy1)
% Found ((and_rect1 ((R x3) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x3) Xy1)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x20)) ((R x3) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x3) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found x6:(Xs Xk)
% Instantiate: Xs:=(R x1):(a->Prop);Xy1:=Xk:a
% Found x6 as proof of ((R x1) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((R x1) Xy1)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy1))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy1)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x1) Xy1)
% Found ((and_rect1 ((R x1) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x1) Xy1)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x30)) ((R x1) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x1) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found x500:=(x50 x4):((R (U Xs)) Xy)
% Found (x50 x4) as proof of ((R x3) Xy)
% Found ((x5 Xy) x4) as proof of ((R x3) Xy)
% Found ((x5 Xy) x4) as proof of ((R x3) Xy)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((R x3) Xy)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((R x3) Xy)
% Found ((and_rect1 ((R x3) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((R x3) Xy)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x20)) ((R x3) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((R x3) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found ((conj20 ((x5 Xy1) x4)) ((x5 Xy0) x4)) as proof of ((and ((R x3) Xy1)) ((R Xy1) Xy0))
% Found (((conj2 ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4)) as proof of ((and ((R x3) Xy1)) ((R Xy1) Xy0))
% Found ((((conj ((R x3) Xy1)) ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4)) as proof of ((and ((R x3) Xy1)) ((R Xy1) Xy0))
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy1)) ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4))) as proof of ((and ((R x3) Xy1)) ((R Xy1) Xy0))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy1)) ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4))) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((and ((R x3) Xy1)) ((R Xy1) Xy0)))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy1)) ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4))) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((and ((R x3) Xy1)) ((R Xy1) Xy0))))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy1)) ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4)))) as proof of ((and ((R x3) Xy1)) ((R Xy1) Xy0))
% Found ((and_rect1 ((and ((R x3) Xy1)) ((R Xy1) Xy0))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy1)) ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4)))) as proof of ((and ((R x3) Xy1)) ((R Xy1) Xy0))
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x20)) ((and ((R x3) Xy1)) ((R Xy1) Xy0))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy1)) ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4)))) as proof of ((and ((R x3) Xy1)) ((R Xy1) Xy0))
% Found x500:=(x50 x4):((R (U Xs)) Xy)
% Found (x50 x4) as proof of ((R x1) Xy)
% Found ((x5 Xy) x4) as proof of ((R x1) Xy)
% Found ((x5 Xy) x4) as proof of ((R x1) Xy)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((R x1) Xy)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((R x1) Xy)
% Found ((and_rect1 ((R x1) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((R x1) Xy)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x30)) ((R x1) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((R x1) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found ((conj20 ((x5 Xy1) x4)) ((x5 Xy0) x4)) as proof of ((and ((R x1) Xy1)) ((R Xy1) Xy0))
% Found (((conj2 ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4)) as proof of ((and ((R x1) Xy1)) ((R Xy1) Xy0))
% Found ((((conj ((R x1) Xy1)) ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4)) as proof of ((and ((R x1) Xy1)) ((R Xy1) Xy0))
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy1)) ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4))) as proof of ((and ((R x1) Xy1)) ((R Xy1) Xy0))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy1)) ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4))) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((and ((R x1) Xy1)) ((R Xy1) Xy0)))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy1)) ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4))) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((and ((R x1) Xy1)) ((R Xy1) Xy0))))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy1)) ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4)))) as proof of ((and ((R x1) Xy1)) ((R Xy1) Xy0))
% Found ((and_rect1 ((and ((R x1) Xy1)) ((R Xy1) Xy0))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy1)) ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4)))) as proof of ((and ((R x1) Xy1)) ((R Xy1) Xy0))
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x30)) ((and ((R x1) Xy1)) ((R Xy1) Xy0))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy1)) ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4)))) as proof of ((and ((R x1) Xy1)) ((R Xy1) Xy0))
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((R x3) Xy1)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy1))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy1)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x3) Xy1)
% Found ((and_rect1 ((R x3) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x3) Xy1)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x20)) ((R x3) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x3) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((R Xy1) Xy0)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R Xy1) Xy0))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R Xy1) Xy0)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R Xy1) Xy0)
% Found ((and_rect1 ((R Xy1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R Xy1) Xy0)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x20)) ((R Xy1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R Xy1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy00:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy00)))
% Found (x50 x4) as proof of ((R x3) Xy00)
% Found ((x5 Xy00) x4) as proof of ((R x3) Xy00)
% Found ((x5 Xy00) x4) as proof of ((R x3) Xy00)
% Found ((x5 Xy00) x4) as proof of ((R x3) Xy00)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy)))
% Found (x50 x4) as proof of ((R Xy00) Xy)
% Found ((x5 Xy) x4) as proof of ((R Xy00) Xy)
% Found ((x5 Xy) x4) as proof of ((R Xy00) Xy)
% Found ((conj20 ((x5 Xy00) x4)) ((x5 Xy) x4)) as proof of ((and ((R x3) Xy00)) ((R Xy00) Xy))
% Found (((conj2 ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) x4)) as proof of ((and ((R x3) Xy00)) ((R Xy00) Xy))
% Found ((((conj ((R x3) Xy00)) ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) x4)) as proof of ((and ((R x3) Xy00)) ((R Xy00) Xy))
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy00)) ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) x4))) as proof of ((and ((R x3) Xy00)) ((R Xy00) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy00)) ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) x4))) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((and ((R x3) Xy00)) ((R Xy00) Xy)))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy00)) ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) x4))) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((and ((R x3) Xy00)) ((R Xy00) Xy))))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy00)) ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) x4)))) as proof of ((and ((R x3) Xy00)) ((R Xy00) Xy))
% Found ((and_rect1 ((and ((R x3) Xy00)) ((R Xy00) Xy))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy00)) ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) x4)))) as proof of ((and ((R x3) Xy00)) ((R Xy00) Xy))
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x20)) ((and ((R x3) Xy00)) ((R Xy00) Xy))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy00)) ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) x4)))) as proof of ((and ((R x3) Xy00)) ((R Xy00) Xy))
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy00:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy00)))
% Found (x50 x4) as proof of ((R x3) Xy00)
% Found ((x5 Xy00) x4) as proof of ((R x3) Xy00)
% Found ((x5 Xy00) x4) as proof of ((R x3) Xy00)
% Found ((x5 Xy00) x4) as proof of ((R x3) Xy00)
% Found x40:=(x4 Xk):((Xs Xk)->((R Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((R Xk) Xy))
% Found (x4 Xk) as proof of ((Xs Xk)->((R Xk) Xy))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of ((Xs Xk)->((R Xk) Xy))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy)))
% Found (x50 (fun (Xk:a)=> (x4 Xk))) as proof of ((R Xy00) Xy)
% Found ((x5 Xy) (fun (Xk:a)=> (x4 Xk))) as proof of ((R Xy00) Xy)
% Found ((x5 Xy) (fun (Xk:a)=> (x4 Xk))) as proof of ((R Xy00) Xy)
% Found ((conj20 ((x5 Xy00) x4)) ((x5 Xy) (fun (Xk:a)=> (x4 Xk)))) as proof of ((and ((R x3) Xy00)) ((R Xy00) Xy))
% Found (((conj2 ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) (fun (Xk:a)=> (x4 Xk)))) as proof of ((and ((R x3) Xy00)) ((R Xy00) Xy))
% Found ((((conj ((R x3) Xy00)) ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) (fun (Xk:a)=> (x4 Xk)))) as proof of ((and ((R x3) Xy00)) ((R Xy00) Xy))
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy00)) ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) (fun (Xk:a)=> (x4 Xk))))) as proof of ((and ((R x3) Xy00)) ((R Xy00) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy00)) ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) (fun (Xk:a)=> (x4 Xk))))) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((and ((R x3) Xy00)) ((R Xy00) Xy)))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy00)) ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) (fun (Xk:a)=> (x4 Xk))))) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((and ((R x3) Xy00)) ((R Xy00) Xy))))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy00)) ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) (fun (Xk:a)=> (x4 Xk)))))) as proof of ((and ((R x3) Xy00)) ((R Xy00) Xy))
% Found ((and_rect1 ((and ((R x3) Xy00)) ((R Xy00) Xy))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy00)) ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) (fun (Xk:a)=> (x4 Xk)))))) as proof of ((and ((R x3) Xy00)) ((R Xy00) Xy))
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x20)) ((and ((R x3) Xy00)) ((R Xy00) Xy))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy00)) ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) (fun (Xk:a)=> (x4 Xk)))))) as proof of ((and ((R x3) Xy00)) ((R Xy00) Xy))
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found x500:=(x50 x4):((R (U Xs)) Xy1)
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((conj20 ((x5 Xy1) x4)) ((x5 Xy0) x4)) as proof of ((and ((R x3) Xy1)) ((R Xy1) Xy0))
% Found (((conj2 ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4)) as proof of ((and ((R x3) Xy1)) ((R Xy1) Xy0))
% Found ((((conj ((R x3) Xy1)) ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4)) as proof of ((and ((R x3) Xy1)) ((R Xy1) Xy0))
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy1)) ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4))) as proof of ((and ((R x3) Xy1)) ((R Xy1) Xy0))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy1)) ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4))) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((and ((R x3) Xy1)) ((R Xy1) Xy0)))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy1)) ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4))) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((and ((R x3) Xy1)) ((R Xy1) Xy0))))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy1)) ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4)))) as proof of ((and ((R x3) Xy1)) ((R Xy1) Xy0))
% Found ((and_rect1 ((and ((R x3) Xy1)) ((R Xy1) Xy0))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy1)) ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4)))) as proof of ((and ((R x3) Xy1)) ((R Xy1) Xy0))
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x20)) ((and ((R x3) Xy1)) ((R Xy1) Xy0))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x3) Xy1)) ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4)))) as proof of ((and ((R x3) Xy1)) ((R Xy1) Xy0))
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((R Xy1) Xy0)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R Xy1) Xy0))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R Xy1) Xy0)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R Xy1) Xy0)
% Found ((and_rect1 ((R Xy1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R Xy1) Xy0)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x20)) ((R Xy1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R Xy1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((R Xy1) Xy0)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R Xy1) Xy0))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R Xy1) Xy0)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R Xy1) Xy0)
% Found ((and_rect1 ((R Xy1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R Xy1) Xy0)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x30)) ((R Xy1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R Xy1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((R x1) Xy1)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy1))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy1)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x1) Xy1)
% Found ((and_rect1 ((R x1) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x1) Xy1)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x30)) ((R x1) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x1) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found x500:=(x50 x4):((R (U Xs)) Xy1)
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((conj20 ((x5 Xy1) x4)) ((x5 Xy0) x4)) as proof of ((and ((R x1) Xy1)) ((R Xy1) Xy0))
% Found (((conj2 ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4)) as proof of ((and ((R x1) Xy1)) ((R Xy1) Xy0))
% Found ((((conj ((R x1) Xy1)) ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4)) as proof of ((and ((R x1) Xy1)) ((R Xy1) Xy0))
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy1)) ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4))) as proof of ((and ((R x1) Xy1)) ((R Xy1) Xy0))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy1)) ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4))) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((and ((R x1) Xy1)) ((R Xy1) Xy0)))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy1)) ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4))) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((and ((R x1) Xy1)) ((R Xy1) Xy0))))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy1)) ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4)))) as proof of ((and ((R x1) Xy1)) ((R Xy1) Xy0))
% Found ((and_rect1 ((and ((R x1) Xy1)) ((R Xy1) Xy0))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy1)) ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4)))) as proof of ((and ((R x1) Xy1)) ((R Xy1) Xy0))
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x30)) ((and ((R x1) Xy1)) ((R Xy1) Xy0))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy1)) ((R Xy1) Xy0)) ((x5 Xy1) x4)) ((x5 Xy0) x4)))) as proof of ((and ((R x1) Xy1)) ((R Xy1) Xy0))
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy00:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy00)))
% Found (x50 x4) as proof of ((R x1) Xy00)
% Found ((x5 Xy00) x4) as proof of ((R x1) Xy00)
% Found ((x5 Xy00) x4) as proof of ((R x1) Xy00)
% Found ((x5 Xy00) x4) as proof of ((R x1) Xy00)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy)))
% Found (x50 x4) as proof of ((R Xy00) Xy)
% Found ((x5 Xy) x4) as proof of ((R Xy00) Xy)
% Found ((x5 Xy) x4) as proof of ((R Xy00) Xy)
% Found ((conj20 ((x5 Xy00) x4)) ((x5 Xy) x4)) as proof of ((and ((R x1) Xy00)) ((R Xy00) Xy))
% Found (((conj2 ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) x4)) as proof of ((and ((R x1) Xy00)) ((R Xy00) Xy))
% Found ((((conj ((R x1) Xy00)) ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) x4)) as proof of ((and ((R x1) Xy00)) ((R Xy00) Xy))
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy00)) ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) x4))) as proof of ((and ((R x1) Xy00)) ((R Xy00) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy00)) ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) x4))) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((and ((R x1) Xy00)) ((R Xy00) Xy)))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy00)) ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) x4))) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((and ((R x1) Xy00)) ((R Xy00) Xy))))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy00)) ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) x4)))) as proof of ((and ((R x1) Xy00)) ((R Xy00) Xy))
% Found ((and_rect1 ((and ((R x1) Xy00)) ((R Xy00) Xy))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy00)) ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) x4)))) as proof of ((and ((R x1) Xy00)) ((R Xy00) Xy))
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x30)) ((and ((R x1) Xy00)) ((R Xy00) Xy))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy00)) ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) x4)))) as proof of ((and ((R x1) Xy00)) ((R Xy00) Xy))
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy00:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy00)))
% Found (x50 x4) as proof of ((R x1) Xy00)
% Found ((x5 Xy00) x4) as proof of ((R x1) Xy00)
% Found ((x5 Xy00) x4) as proof of ((R x1) Xy00)
% Found ((x5 Xy00) x4) as proof of ((R x1) Xy00)
% Found x40:=(x4 Xk):((Xs Xk)->((R Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((R Xk) Xy))
% Found (x4 Xk) as proof of ((Xs Xk)->((R Xk) Xy))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of ((Xs Xk)->((R Xk) Xy))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy)))
% Found (x50 (fun (Xk:a)=> (x4 Xk))) as proof of ((R Xy00) Xy)
% Found ((x5 Xy) (fun (Xk:a)=> (x4 Xk))) as proof of ((R Xy00) Xy)
% Found ((x5 Xy) (fun (Xk:a)=> (x4 Xk))) as proof of ((R Xy00) Xy)
% Found ((conj20 ((x5 Xy00) x4)) ((x5 Xy) (fun (Xk:a)=> (x4 Xk)))) as proof of ((and ((R x1) Xy00)) ((R Xy00) Xy))
% Found (((conj2 ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) (fun (Xk:a)=> (x4 Xk)))) as proof of ((and ((R x1) Xy00)) ((R Xy00) Xy))
% Found ((((conj ((R x1) Xy00)) ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) (fun (Xk:a)=> (x4 Xk)))) as proof of ((and ((R x1) Xy00)) ((R Xy00) Xy))
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy00)) ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) (fun (Xk:a)=> (x4 Xk))))) as proof of ((and ((R x1) Xy00)) ((R Xy00) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy00)) ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) (fun (Xk:a)=> (x4 Xk))))) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((and ((R x1) Xy00)) ((R Xy00) Xy)))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy00)) ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) (fun (Xk:a)=> (x4 Xk))))) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((and ((R x1) Xy00)) ((R Xy00) Xy))))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy00)) ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) (fun (Xk:a)=> (x4 Xk)))))) as proof of ((and ((R x1) Xy00)) ((R Xy00) Xy))
% Found ((and_rect1 ((and ((R x1) Xy00)) ((R Xy00) Xy))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy00)) ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) (fun (Xk:a)=> (x4 Xk)))))) as proof of ((and ((R x1) Xy00)) ((R Xy00) Xy))
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x30)) ((and ((R x1) Xy00)) ((R Xy00) Xy))) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((((conj ((R x1) Xy00)) ((R Xy00) Xy)) ((x5 Xy00) x4)) ((x5 Xy) (fun (Xk:a)=> (x4 Xk)))))) as proof of ((and ((R x1) Xy00)) ((R Xy00) Xy))
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((R x3) Xy1)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy1))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy1)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x3) Xy1)
% Found ((and_rect1 ((R x3) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x3) Xy1)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x20)) ((R x3) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x3) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((R x3) Xy1)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy1))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy1)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x3) Xy1)
% Found ((and_rect1 ((R x3) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x3) Xy1)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x20)) ((R x3) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x3) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((R Xy1) Xy0)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R Xy1) Xy0))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R Xy1) Xy0)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R Xy1) Xy0)
% Found ((and_rect1 ((R Xy1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R Xy1) Xy0)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x30)) ((R Xy1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R Xy1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((R x1) Xy1)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy1))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy1)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x1) Xy1)
% Found ((and_rect1 ((R x1) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x1) Xy1)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x30)) ((R x1) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x1) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((R x1) Xy1)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy1))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy1)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x1) Xy1)
% Found ((and_rect1 ((R x1) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x1) Xy1)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x30)) ((R x1) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x1) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((R x3) Xy1)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy1))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy1)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x3) Xy1)
% Found ((and_rect1 ((R x3) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x3) Xy1)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x20)) ((R x3) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x3) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((R Xy1) Xy0)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R Xy1) Xy0))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R Xy1) Xy0)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R Xy1) Xy0)
% Found ((and_rect1 ((R Xy1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R Xy1) Xy0)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x20)) ((R Xy1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R Xy1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((R x1) Xy1)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy1))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy1)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x1) Xy1)
% Found ((and_rect1 ((R x1) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x1) Xy1)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x30)) ((R x1) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x1) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((R Xy1) Xy0)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R Xy1) Xy0))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R Xy1) Xy0)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R Xy1) Xy0)
% Found ((and_rect1 ((R Xy1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R Xy1) Xy0)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x30)) ((R Xy1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R Xy1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy00:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy00)))
% Found (x50 x4) as proof of ((R x3) Xy00)
% Found ((x5 Xy00) x4) as proof of ((R x3) Xy00)
% Found ((x5 Xy00) x4) as proof of ((R x3) Xy00)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy00) x4)) as proof of ((R x3) Xy00)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy00) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy00))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy00) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy00)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy00) x4))) as proof of ((R x3) Xy00)
% Found ((and_rect1 ((R x3) Xy00)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy00) x4))) as proof of ((R x3) Xy00)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x20)) ((R x3) Xy00)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy00) x4))) as proof of ((R x3) Xy00)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy)))
% Found (x50 x4) as proof of ((R Xy00) Xy)
% Found ((x5 Xy) x4) as proof of ((R Xy00) Xy)
% Found ((x5 Xy) x4) as proof of ((R Xy00) Xy)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((R Xy00) Xy)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R Xy00) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R Xy00) Xy)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((R Xy00) Xy)
% Found ((and_rect1 ((R Xy00) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((R Xy00) Xy)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x20)) ((R Xy00) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((R Xy00) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy00:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy00)))
% Found (x50 x4) as proof of ((R x3) Xy00)
% Found ((x5 Xy00) x4) as proof of ((R x3) Xy00)
% Found ((x5 Xy00) x4) as proof of ((R x3) Xy00)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy00) x4)) as proof of ((R x3) Xy00)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy00) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy00))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy00) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy00)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy00) x4))) as proof of ((R x3) Xy00)
% Found ((and_rect1 ((R x3) Xy00)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy00) x4))) as proof of ((R x3) Xy00)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x20)) ((R x3) Xy00)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy00) x4))) as proof of ((R x3) Xy00)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((R x3) Xy1)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy1))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy1)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x3) Xy1)
% Found ((and_rect1 ((R x3) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x3) Xy1)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x20)) ((R x3) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x3) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((R Xy1) Xy0)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R Xy1) Xy0))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R Xy1) Xy0)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R Xy1) Xy0)
% Found ((and_rect1 ((R Xy1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R Xy1) Xy0)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x20)) ((R Xy1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R Xy1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((R x3) Xy1)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy1))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy1)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x3) Xy1)
% Found ((and_rect1 ((R x3) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x3) Xy1)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x20)) ((R x3) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x3) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy)))
% Found (x50 x4) as proof of ((R Xy00) Xy)
% Found ((x5 Xy) x4) as proof of ((R Xy00) Xy)
% Found ((x5 Xy) x4) as proof of ((R Xy00) Xy)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((R Xy00) Xy)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R Xy00) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R Xy00) Xy)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((R Xy00) Xy)
% Found ((and_rect1 ((R Xy00) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((R Xy00) Xy)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x30)) ((R Xy00) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) x4))) as proof of ((R Xy00) Xy)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy00:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy00)))
% Found (x50 x4) as proof of ((R x1) Xy00)
% Found ((x5 Xy00) x4) as proof of ((R x1) Xy00)
% Found ((x5 Xy00) x4) as proof of ((R x1) Xy00)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy00) x4)) as proof of ((R x1) Xy00)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy00) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy00))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy00) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy00)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy00) x4))) as proof of ((R x1) Xy00)
% Found ((and_rect1 ((R x1) Xy00)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy00) x4))) as proof of ((R x1) Xy00)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x30)) ((R x1) Xy00)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy00) x4))) as proof of ((R x1) Xy00)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((R x1) Xy1)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy1))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy1)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x1) Xy1)
% Found ((and_rect1 ((R x1) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x1) Xy1)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x30)) ((R x1) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x1) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy0:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy0)))
% Found (x50 x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found ((x5 Xy0) x4) as proof of ((R Xy1) Xy0)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((R Xy1) Xy0)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R Xy1) Xy0))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R Xy1) Xy0)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R Xy1) Xy0)
% Found ((and_rect1 ((R Xy1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R Xy1) Xy0)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x30)) ((R Xy1) Xy0)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy0) x4))) as proof of ((R Xy1) Xy0)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy00:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy00)))
% Found (x50 x4) as proof of ((R x1) Xy00)
% Found ((x5 Xy00) x4) as proof of ((R x1) Xy00)
% Found ((x5 Xy00) x4) as proof of ((R x1) Xy00)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy00) x4)) as proof of ((R x1) Xy00)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy00) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy00))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy00) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy00)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy00) x4))) as proof of ((R x1) Xy00)
% Found ((and_rect1 ((R x1) Xy00)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy00) x4))) as proof of ((R x1) Xy00)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x30)) ((R x1) Xy00)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy00) x4))) as proof of ((R x1) Xy00)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((R x1) Xy1)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy1))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy1)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x1) Xy1)
% Found ((and_rect1 ((R x1) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x1) Xy1)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x30)) ((R x1) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x1) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x3) Xy1)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((R x3) Xy1)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy1))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x3) Xy1)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x3) Xy1)
% Found ((and_rect1 ((R x3) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x3) Xy1)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x20)) ((R x3) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x3) Xy1)
% Found x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))
% Instantiate: Xy1:=(U Xs):a
% Found x4 as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy1)))
% Found (x50 x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found ((x5 Xy1) x4) as proof of ((R x1) Xy1)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((R x1) Xy1)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy1))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4)) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R x1) Xy1)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x1) Xy1)
% Found ((and_rect1 ((R x1) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x1) Xy1)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x30)) ((R x1) Xy1)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy1) x4))) as proof of ((R x1) Xy1)
% Found x40:=(x4 Xk):((Xs Xk)->((R Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((R Xk) Xy))
% Found (x4 Xk) as proof of ((Xs Xk)->((R Xk) Xy))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of ((Xs Xk)->((R Xk) Xy))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy)))
% Found (x50 (fun (Xk:a)=> (x4 Xk))) as proof of ((R Xy00) Xy)
% Found ((x5 Xy) (fun (Xk:a)=> (x4 Xk))) as proof of ((R Xy00) Xy)
% Found ((x5 Xy) (fun (Xk:a)=> (x4 Xk))) as proof of ((R Xy00) Xy)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) (fun (Xk:a)=> (x4 Xk)))) as proof of ((R Xy00) Xy)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) (fun (Xk:a)=> (x4 Xk)))) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R Xy00) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) (fun (Xk:a)=> (x4 Xk)))) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R Xy00) Xy)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) (fun (Xk:a)=> (x4 Xk))))) as proof of ((R Xy00) Xy)
% Found ((and_rect1 ((R Xy00) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) (fun (Xk:a)=> (x4 Xk))))) as proof of ((R Xy00) Xy)
% Found (((fun (P:Type) (x4:((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->P)))=> (((((and_rect (forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))) P) x4) x20)) ((R Xy00) Xy)) (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) (fun (Xk:a)=> (x4 Xk))))) as proof of ((R Xy00) Xy)
% Found x40:=(x4 Xk):((Xs Xk)->((R Xk) (U Xs)))
% Found (x4 Xk) as proof of ((Xs Xk)->((R Xk) Xy))
% Found (x4 Xk) as proof of ((Xs Xk)->((R Xk) Xy))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of ((Xs Xk)->((R Xk) Xy))
% Found (fun (Xk:a)=> (x4 Xk)) as proof of (forall (Xk:a), ((Xs Xk)->((R Xk) Xy)))
% Found (x50 (fun (Xk:a)=> (x4 Xk))) as proof of ((R Xy00) Xy)
% Found ((x5 Xy) (fun (Xk:a)=> (x4 Xk))) as proof of ((R Xy00) Xy)
% Found ((x5 Xy) (fun (Xk:a)=> (x4 Xk))) as proof of ((R Xy00) Xy)
% Found (fun (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) (fun (Xk:a)=> (x4 Xk)))) as proof of ((R Xy00) Xy)
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) (fun (Xk:a)=> (x4 Xk)))) as proof of ((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R Xy00) Xy))
% Found (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) (fun (Xk:a)=> (x4 Xk)))) as proof of ((forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))->((forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj)))->((R Xy00) Xy)))
% Found (and_rect10 (fun (x4:(forall (Xz:a), ((Xs Xz)->((R Xz) (U Xs))))) (x5:(forall (Xj:a), ((forall (Xk:a), ((Xs Xk)->((R Xk) Xj)))->((R (U Xs)) Xj))))=> ((x5 Xy) (fun (Xk:a)=> (x4 Xk))))) as proof of ((R Xy00) Xy)
% Found ((and_rect1 ((R Xy00) Xy)) (fun (x4:(f
% EOF
%------------------------------------------------------------------------------