
theorem
for R being preordered domRing,
    P being Preordering of R,
    a being P-ordered Element of R holds abs(P,a)^2 = a^2
proof
let R be preordered domRing, P be Preordering of R,
    a be P-ordered Element of R;
a in P\/-P by defppp; then
per cases by XBOOLE_0:def 3;
suppose a in P;
  then 0.R <=P, a;
  hence thesis by av2;
  end;
suppose a in -P;
  then -a in --P;
  then a <=P, 0.R;
  hence abs(P,a)^2 = (-a)^2 by av3
                  .= a^2 by VECTSP_1:10;
  end;
end;
