
theorem
for O being Ordering of F_Real holds O = Positives(F_Real)
proof
let O be Ordering of F_Real;
X: QS F_Real c= O by ord2;
now let x be object;
  assume x in Positives(F_Real);
  then consider r being Element of F_Real such that A: x = r & 0 <= r;
  reconsider q = sqrt r as Element of F_Real by XREAL_0:def 1;
  r = (sqrt r)^2 by A,SQUARE_1:def 2
   .= q^2;
  then r is square;
  then r in QS F_Real;
  hence x in O by A,X;
  end;
then Positives(F_Real) c= O;
hence thesis by ordsub;
end;
