let h be non constant standard special_circular_sequence; :: thesis: E-bound (L~ h) = ((GoB h) * (len (GoB h)),1) `1
set X = { (q `1 ) where q is Point of (TOP-REAL 2) : q in L~ h } ;
set A = ((GoB h) * (len (GoB h)),1) `1 ;
consider a being set such that
A1: a in L~ h by XBOOLE_0:def 1;
reconsider a = a as Point of (TOP-REAL 2) by A1;
A2: a `1 in { (q `1 ) where q is Point of (TOP-REAL 2) : q in L~ h } by A1;
{ (q `1 ) where q is Point of (TOP-REAL 2) : q in L~ h } c= REAL

proof

let b be set ; :: according to TARSKI:def 3 :: thesis: ( not b in { (q `1 ) where q is Point of (TOP-REAL 2) : q in L~ h } or b in REAL )
assume b in { (q `1 ) where q is Point of (TOP-REAL 2) : q in L~ h } ; :: thesis: b in REAL
then consider qq being Point of (TOP-REAL 2) such that
A3: ( b = qq `1 & qq in L~ h ) ;
thus b in REAL by A3; :: thesis: verum

end;

then reconsider X = { (q `1 ) where q is Point of (TOP-REAL 2) : q in L~ h } as non empty Subset of REAL by A2;
upper_bound X = ((GoB h) * (len (GoB h)),1) `1

proof

A4: for p being real number st p in X holds
p <= ((GoB h) * (len (GoB h)),1) `1

proof

let p be real number ; :: thesis: ( p in X implies p <= ((GoB h) * (len (GoB h)),1) `1 )
assume p in X ; :: thesis: p <= ((GoB h) * (len (GoB h)),1) `1
then consider s being Point of (TOP-REAL 2) such that
A5: ( p = s `1 & s in L~ h ) ;
1 <= width (GoB h) by GOBOARD7:35;
hence p <= ((GoB h) * (len (GoB h)),1) `1 by A5, Th34; :: thesis: verum

end;

( 1 <= len (GoB h) & 1 <= width (GoB h) ) by GOBOARD7:34, GOBOARD7:35;
then consider q1 being Point of (TOP-REAL 2) such that
A6: ( q1 `1 = ((GoB h) * (len (GoB h)),1) `1 & q1 in L~ h ) by Th37;
reconsider q11 = q1 `1 as Real ;
for q being real number st ( for p being real number st p in X holds
p <= q ) holds
((GoB h) * (len (GoB h)),1) `1 <= q

proof

let q be real number ; :: thesis: ( ( for p being real number st p in X holds
p <= q ) implies ((GoB h) * (len (GoB h)),1) `1 <= q )
assume A7: for p being real number st p in X holds
p <= q ; :: thesis: ((GoB h) * (len (GoB h)),1) `1 <= q
q11 in X by A6;
hence ((GoB h) * (len (GoB h)),1) `1 <= q by A6, A7; :: thesis: verum

end;

hence upper_bound X = ((GoB h) * (len (GoB h)),1) `1 by A4, SEQ_4:63; :: thesis: verum

end;

hence E-bound (L~ h) = ((GoB h) * (len (GoB h)),1) `1 by Th19; :: thesis: verum