
theorem T1:
for F being preordered Field,
    P being Preordering of F holds P is maximal iff P is positive_cone
proof
let R be preordered Field, P be Preordering of R;
hereby assume AS: P is maximal;
  now let x be object;
    assume x in the carrier of R;
    then reconsider a = x as Element of R;
    now assume AS1: not(a in P);
      now assume not(-a in P);
        then reconsider Q = P + a * P as Preordering of R by T2;
        C: 0.R in P by REALALG1:25;
        then X: P = Q by AS,P1;
        1.R in P by REALALG1:25;
        hence contradiction by P2,C,X,AS1;
        end;
      then --a in -P;
      hence a in -P;
      end;
    hence x in P \/ (-P) by XBOOLE_0:def 3;
    end;
  then the carrier of R c= P \/ (-P);
  then P \/ (-P) = the carrier of R;
  then P is spanning;
  hence P is positive_cone;
  end;
assume AS: P is positive_cone;
   assume not P is maximal;
     then consider Q being Preordering of R such that A: P c= Q & P <> Q;
     P c< Q by A; then
     consider a being object such that B: a in Q & not(a in P) by XBOOLE_0:6;
     reconsider a as Element of R by B;
     a in (-P) by AS,B,XBOOLE_0:def 3;
     then -a in --P;
     then --a in -Q by A;
     then a in Q /\ (-Q) by B;
     then a in {0.R} by REALALG1:def 7;
     then a = 0.R by TARSKI:def 1;
     hence contradiction by B,REALALG1:25;
end;
