theorem Th54:
  p`1 <= q`1 implies W-bound LSeg(p,q) = p`1
proof
  assume
A1: p`1 <= q`1;
  then
A2: proj1.:LSeg(p,q) = [.p`1,q`1.] by Th52;
  thus W-bound LSeg(p,q) = lower_bound(proj1.:LSeg(p,q)) by Th43
    .= p`1 by A1,A2,JORDAN5A:19;
end;
