
theorem
for R being preordered Ring
for P being Preordering of R
for S being Subring of R holds P /\ (the carrier of S) is Preordering of S
proof
let R be preordered Ring, O be Preordering 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 prepositive_cone by lemsubpreord;
hence thesis;
end;
