reserve p, q for Point of TOP-REAL 2,
  r for Real,
  h for non constant standard special_circular_sequence,
  g for FinSequence of TOP-REAL 2,
  f for non empty FinSequence of TOP-REAL 2,
  I, i1, i, j, k for Nat;

theorem Th40:
  N-bound L~h = (GoB h)*(1,width GoB h)`2
proof
  set X = { q`2 : q in L~h }, A = (GoB h)*(1,width GoB h)`2;
  consider a be object such that
A1: a in L~h by XBOOLE_0:def 1;
A2: X c= REAL
  proof
    let b be object;
    assume b in X;
    then ex qq be Point of TOP-REAL 2 st b = qq`2 & qq in L~h;
    hence thesis by XREAL_0:def 1;
  end;
  reconsider a as Point of TOP-REAL 2 by A1;
  a`2 in X by A1;
  then reconsider X as non empty Subset of REAL by A2;
  upper_bound X = A
  proof
A3: 1 <= len GoB h by GOBOARD7:32;
A4: for p be Real st p in X holds p <= A
    proof
      let p be Real;
      assume p in X;
      then ex s be Point of TOP-REAL 2 st p = s`2 & s in L~h;
      hence thesis by A3,Th34;
    end;
    1 <= width GoB h by GOBOARD7:33;
    then consider q1 be Point of TOP-REAL 2 such that
A5: q1`2 = (GoB h)*(1,width GoB h)`2 and
A6: q1 in L~h by A3,Th36;
    reconsider q11 = q1`2 as Real;
    for q be Real st for p be Real st p in X holds p <= q
    holds A <= q
    proof
A7:   q11 in X by A6;
      let q be Real;
      assume for p be Real st p in X holds p <= q;
      hence thesis by A5,A7;
    end;
    hence thesis by A4,SEQ_4:46;
  end;
  hence thesis by Th18;
end;
