
theorem
for F being ordered Field,
    E being ordered FieldExtension of F
for P being Ordering of F,
    O being Ordering of E
holds O extends P iff (signum O)|(the carrier of F) = signum P
proof
let F be ordered Field, E be ordered FieldExtension of F,
    P be Ordering of F, O be Ordering of E;
set sP = signum P, sO = signum O;
H: F is Subfield of E &
   P \/ -P = the carrier of F by REALALG1:def 15,FIELD_4:7; then
I: the carrier of F c= the carrier of E by EC_PF_1:def 1;
C: dom sO /\ (the carrier of F)
     = (the carrier of E) /\ (the carrier of F) by FUNCT_2:def 1
    .= the carrier of F by I,XBOOLE_1:28
    .= dom sP by FUNCT_2:def 1;
A: now assume B: O extends P;
   now let x be object;
     assume x in dom sP; then
     reconsider a = x as Element of F by FUNCT_2:def 1;
     reconsider b = a as Element of E by I;
     per cases;
     suppose E: x in P \ {0.F}; then
       a in P & not a in {0.F} by XBOOLE_0:def 5; then
       a in O & not a in {0.E} by B,H,XBOOLE_0:def 4,EC_PF_1:def 1;
       then F: x in O \ {0.E} by XBOOLE_0:def 5;
       thus sO.x = signum(O,b) by sgnF
                .= 1 by F,defsgn
                .= signum(P,a) by E,defsgn
                .= sP.x by sgnF;
       end;
     suppose E: x = 0.F; then
       F: x = 0.E by H,EC_PF_1:def 1;
       thus sO.x = signum(O,b) by sgnF
                .= 0 by F,defsgn
                .= signum(P,a) by E,defsgn
                .= sP.x by sgnF;
       end;
     suppose E: not x in P \ {0.F} & x <> 0.F; then
       not a in P or a in {0.F} by XBOOLE_0:def 5; then
       not a in O by B,XBOOLE_0:def 4,E,TARSKI:def 1; then
       F: not x in O \ {0.E} & x <> 0.E by H,E,EC_PF_1:def 1,XBOOLE_0:def 5;
       thus sO.x = signum(O,b) by sgnF
                .= -1 by F,defsgn
                .= signum(P,a) by E,defsgn
                .= sP.x by sgnF;
       end;
     end;
   hence (signum O)|(the carrier of F) = signum P by C,FUNCT_1:46;
   end;
now assume A: (signum O)|(the carrier of F) = signum P;
  now let x be object;
    assume B: x in P;
    per cases;
    suppose x <> 0.F; then
      not x in {0.F} by TARSKI:def 1; then
      D: x in P \ {0.F} by B,XBOOLE_0:def 5;
      reconsider a = x as Element of F by B;
      reconsider b = a as Element of E by I;
      E: dom sP = the carrier of F by FUNCT_2:def 1;
      signum(O,b) = sO.b by sgnF
          .= sP.a by C,E,A,FUNCT_1:48
          .= signum(P,a) by sgnF
          .= 1 by D,defsgn;
      then b in O \ {0.E} by defsgn;
      hence x in O by XBOOLE_0:def 5;
      end;
    suppose x = 0.F;
      then x = 0.E by H,EC_PF_1:def 1;
      hence x in O by REALALG1:25;
      end;
    end;
  then P c= O;
  hence O extends P by l13;
  end;
hence thesis by A;
end;
