
theorem
for R being ordered Ring
for O being Ordering of R
for S being Subring of R holds O /\ (the carrier of S) is Ordering of S
proof
let R be ordered Ring, O be Ordering of R, S be Subring of R;
for o be object st o in O /\ (the carrier of S) holds
  o in the carrier of S by XBOOLE_0:def 4;
then reconsider M = O /\ (the carrier of S) as Subset of S by TARSKI:def 3;
M is positive_cone by lemsubord;
hence thesis;
end;
