
theorem ord4:
for R being preordered non degenerated Ring
for P being Preordering of R holds not -1.R in P
proof
let R be preordered non degenerated Ring, P be Preordering of R;
A: 1.R in P by ord3;
assume -1.R in P;
then -(-1.R) in - P;
then 1.R in P /\ -P by XBOOLE_0:def 4,A;
then 1.R in {0.R} by defppc;
hence contradiction by TARSKI:def 1;
end;
