theorem Th57:
  p`1 <= q`1 implies E-bound LSeg(p,q) = q`1
proof
  assume
A1: p`1 <= q`1;
  then
A2: proj1.:LSeg(p,q) = [.p`1,q`1.] by Th52;
  thus E-bound LSeg(p,q) = upper_bound(proj1.:LSeg(p,q)) by Th46
    .= q`1 by A1,A2,JORDAN5A:19;
end;
