
theorem c55:
for R being preordered Ring,
    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 Ring, 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);
then A: a * (-c) <=P, b * (-c) by AS,c5;
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;
end;
