
theorem
for R being preordered domRing,
    P being Preordering of R
holds P^- * (P^-) c= P^+ & P^+ * (P^-) c= P^- & P^- * (P^+) c= P^-
proof
let R be preordered domRing; let P be Preordering of R;
now 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: a in -P & not a in {0.R} &
   b in -P & not b in {0.R} by A,XBOOLE_0:def 5; then
C: a <> 0.R & b <> 0.R by TARSKI:def 1;
D: (-P) * (-P) c= P & o in (-P) * (-P) by A,B,REALALG2:17;
   now assume a * b in {0.R};
     then a * b = 0.R by TARSKI:def 1;
     hence contradiction by C,VECTSP_2:def 1;
     end;
   hence o in P^+ by D,A,XBOOLE_0:def 5;
  end;
hence P^- * (P^-) c= P^+;
now 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: a in P & not a in {0.R} &
   b in -P & not b in {0.R} by A,XBOOLE_0:def 5; then
C: a <> 0.R & b <> 0.R by TARSKI:def 1;
D: P * (-P) c= -P & o in P * (-P) by A,B,REALALG2:18;
   now assume a * b in {0.R};
     then a * b = 0.R by TARSKI:def 1;
     hence contradiction by C,VECTSP_2:def 1;
     end;
   hence o in P^- by D,A,XBOOLE_0:def 5;
  end;
hence P^+ * (P^-) c= P^-;
now 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: a in -P & not a in {0.R} &
   b in P & not b in {0.R} by A,XBOOLE_0:def 5; then
C: a <> 0.R & b <> 0.R by TARSKI:def 1;
D: (-P) * P c= -P & o in (-P) * P by A,B,REALALG2:18;
   now assume a * b in {0.R};
     then a * b = 0.R by TARSKI:def 1;
     hence contradiction by C,VECTSP_2:def 1;
     end;
   hence o in P^- by D,A,XBOOLE_0:def 5;
  end;
hence P^- * (P^+) c= P^-;
end;
