
theorem
for R being ordered domRing,
    O being Ordering of R,
    a,b being non O-negative Element of R holds a <O, b iff a^2 <O, b^2
proof
let R be ordered domRing, P be Ordering of R,
    a,b be non P-negative Element of R;
the carrier of R = P \/ -P by REALALG1:def 8;
then a is P-ordered & b is P-ordered;
then AS: 0.R <=P, a & 0.R <=P, b by x1a;
hereby assume K: a <P, b;
  a is Sqrt of a^2 & b is Sqrt of b^2 by defsqrt;
  hence a^2 <P, b^2 by K,AS,sq1,sq0;
  end;
thus thesis by sq1;
end;
