
theorem avb4:
for R being ordered Ring,
    O being Ordering of R,
    a,b being Element of R holds a <=O, b or b <=O ,a
proof
let R be ordered Ring, O be Ordering of R, a,b be Element of R;
X: O \/ -O = the carrier of R by REALALG1:def 8;
assume not(a <=O, b);
then b - a in -O by X,XBOOLE_0:def 3;
then -(b - a) in --O;
hence b <=O, a by RLVECT_1:33;
end;
