
theorem l13:
for F being ordered Field,
    E being ordered FieldExtension of F
for P being Ordering of F, O be Ordering of E
holds O extends P iff P c= O
proof
let F be ordered Field, E be ordered FieldExtension of F;
let P be Ordering of F, O be Ordering of E;
H1: P \/ (-P) = the carrier of F by REALALG1:def 8;
H2: O /\ (-O) = {0.E} by REALALG1:def 7;
I1: F is Subfield of E & F is Subring of E by FIELD_4:def 1,FIELD_4:7; then
I2: the carrier of F c= the carrier of E & 0.F = 0.E by EC_PF_1:def 1;
now assume AS: P c= O;
  now let a be Element of F;
     reconsider b = a as Element of E by I2;
     I3: -b = -a by I1,FIELD_6:17;
     assume B0: a in O;
     now assume B1: not a in P;
       then a in -P by H1,XBOOLE_0:def 3;
       then -a in --P;
       then --b in -O by AS,I3;
       then b in O /\ (-O) by B0;
       then b = 0.F by I2,H2,TARSKI:def 1;
       hence contradiction by B1,REALALG1:25;
       end;
     hence a in P;
     end;
  hence for a being Element of F holds a in P iff a in O by AS;
  end;
hence thesis by XBOOLE_0:def 4,l12;
end;
