
theorem v2:
for R being preordered Ring,
    P being Preordering of R holds (-P) * P c= -P & P * (-P) c= -P
proof
let R be preordered Ring, P be Preordering of R;
hereby let o be object;
  assume o in (-P) * P;
  then consider a,b being Element of R such that
  A: o = a * b & a in -P & b in P;
  -b in -P by A;
  then B: a * (-b) in (-P) * (-P) by A;
  (-P) * (-P) c= P by v1;
  then a * (-b) in P by B;
  then -(a * b) in P by VECTSP_1:8;
  then --(a * b) in -P;
  hence o in -P by A;
  end; then
(-P) * P c= -P;
hence thesis by v1a;
end;
