
theorem ordsub:
for R being ordered Ring
for O,P being Ordering of R st O c= P holds O = P
proof
let R be ordered Ring, O,P be Ordering of R;
assume AS: O c= P;
now assume not(P c= O);
  then consider a being Element of R such that
  A: a in P & not(a in O);
  a in the carrier of R;
  then a in O \/ - O by defsp;
  then a in -O by A,XBOOLE_0:def 3;
  then B: -a in --O;
  -a in -P by A;
  then -a in P /\ (-P) by AS,B,XBOOLE_0:def 4;
  then -a in {0.R} by defneg;
  then C: -a = 0.R by TARSKI:def 1;
  a = --a .= 0.R by C;
  hence contradiction by A,ord3;
  end;
hence thesis by AS;
end;
