reserve x,y,z for Real,
  R for real non empty RelStr,
  a,b for Element of R;

theorem Th6:
  (ex x st x in the carrier of R & for y st y in the carrier of R
  holds x <= y) iff R is lower-bounded
proof
  hereby
    given x such that
A1: x in the carrier of R and
A2: for y st y in the carrier of R holds x <= y;
    reconsider a = x as Element of R by A1;
    for b st b in the carrier of R holds a <<= b by A2,Th3;
    then a is_<=_than the carrier of R;
    hence R is lower-bounded;
  end;
  given x being Element of R such that
A3: x is_<=_than the carrier of R;
  take x;
  thus x in the carrier of R;
  let y;
  assume y in the carrier of R;
  then reconsider b = y as Element of R;
  x <<= b by A3;
  hence thesis by Th3;
end;
