
theorem
for R being preordered domRing,
    P being Preordering of R,
    a,b,c being Element of R st a <P, b & c <P, 0.R holds b * c <P, a * c
proof
let R be preordered domRing, P be Preordering of R, a,b,c be Element of R;
assume AS: a <P, b & c <P, 0.R;
then -0.R <P, (-c) by c10a,RLVECT_1:18;
then A: a * (-c) <P, b * (-c) by AS,c5,GCD_1:1;
B: -(b * (-c)) = (-b) * (-c) by VECTSP_1:9 .= b * c by VECTSP_1:10;
-(a * (-c)) = (-a) * (-c) by VECTSP_1:9 .= a * c by VECTSP_1:10;
hence thesis by A,B,c10a,RLVECT_1:18;
end;
