
theorem T3:
for F being preordered Field,
    P being Preordering of F
ex Q being Preordering of F st P c= Q & Q is maximal
proof
let F be preordered Field, P be Preordering of F;
set S = {O where O is Preordering of F: P c= O};
set R = RelIncl S;
A2: field R = S &
    for Y,Z being set st Y in S & Z in S holds [Y,Z] in R iff Y c= Z
      by WELLORD2:def 1;
A3: S has_upper_Zorn_property_wrt R
    proof
    now let Y be set;
    assume AS: Y c= S & R|_2 Y is being_linear-order;
    H1: now let z be set;
        assume z in S;
        then consider p being Preordering of F such that H: z = p & P c= p;
        thus P c= z & z is Preordering of F by H;
        end;
    H2: P in S & R |_2 Y = R /\ [:Y,Y:];
    H3: R|_2 Y is connected by AS,ORDERS_1:def 6;
    H5: now let z1,z2 be set;
        assume HH0: z1 in Y & z2 in Y;
        per cases;
        suppose z1 = z2;
          hence z1 c= z2 or z2 c= z1;
          end;
        suppose HH1: z1 <> z2;
          z1 in field(R|_2 Y) & z2 in field(R|_2 Y) by HH0,A2,AS,ORDERS_1:71;
          then [z1,z2] in (R|_2 Y) or [z2,z1] in (R|_2 Y)
                                       by H3,HH1,RELAT_2:def 6,RELAT_2:def 14;
          then [z1,z2] in R or [z2,z1] in R by XBOOLE_0:def 4;
          hence z1 c= z2 or z2 c= z1 by HH0,AS,WELLORD2:def 1;
          end;
        end;
  set M = union Y;
  per cases;
  suppose Y = {};
    then for y being set st y in Y holds [y,P] in R;
    hence ex x being set st x in S &
                         for y being set st y in Y holds [y,x] in R by H2;
    end;
  suppose H: Y <> {};
  A7: M c= the carrier of F
  proof let o be object;
    assume o in M;
    then consider s being set such that H: o in s & s in Y by TARSKI:def 4;
    s is Preordering of F by H1,H,AS;
    hence o in the carrier of F by H;
    end;
  A8a: ex s being set st 0.F in s & s in Y
       proof
       set s = the Element of Y;
       s in Y by H;
       then s is Preordering of F by AS,H1;
       hence thesis by H,REALALG1:25;
       end;
  then A8: 0.F in M by TARSKI:def 4;
  reconsider M as non empty Subset of F by A7,A8a,TARSKI:def 4;
  A6: M is Preordering of F
      proof
      A10: M + M c= M
           proof
           let o be object;
           assume o in M + M;
           then consider a,b being Element of F such that
           A11: o = a + b & a in M & b in M;
           consider sa being set such that A12: a in sa & sa in Y
              by A11,TARSKI:def 4;
           consider sb being set such that A13: b in sb & sb in Y
              by A11,TARSKI:def 4;
           reconsider sa,sb as Preordering of F by A12,A13,AS,H1;
           A16: sa + sa c= sa & sb + sb c= sb by REALALG1:def 14;
           per cases by A12,A13,H5;
           suppose sa c= sb;
             then a + b in sb + sb by A12,A13;
             hence o in M by A16,A11,A13,TARSKI:def 4;
             end;
           suppose sb c= sa;
             then a + b in sa + sa by A12,A13;
             hence o in M by A16,A11,A12,TARSKI:def 4;
             end;
           end;
      A11: M * M c= M
           proof
           let o be object;
           assume o in M * M;
           then consider a,b being Element of F such that
           A11: o = a * b & a in M & b in M;
           consider sa being set such that A12: a in sa & sa in Y
              by A11,TARSKI:def 4;
           consider sb being set such that A13: b in sb & sb in Y
              by A11,TARSKI:def 4;
           reconsider sa,sb as Preordering of F by A12,A13,AS,H1;
           A16: sa * sa c= sa & sb * sb c= sb by REALALG1:def 14;
           per cases by A12,A13,H5;
           suppose sa c= sb;
             then a * b in sb * sb by A12,A13;
             hence o in M by A11,A13,A16,TARSKI:def 4;
             end;
           suppose sb c= sa;
             then a * b in sa * sa by A12,A13;
             hence o in M by A16,A11,A12,TARSKI:def 4;
             end;
           end;
      A12: M /\ (-M) = {0.F}
           proof
           A13: now let o be object;
                assume o in {0.F};
                then A14: o = -0.F by TARSKI:def 1;
                then o in -M by A8;
                hence o in M /\ - M by A14,A8;
                end;
           now let o be object;
             assume A14: o in M /\ - M;
             then A14a: o in M & o in -M by XBOOLE_0:def 4;
             then consider so being set such that A15: o in so & so in Y
                  by TARSKI:def 4;
             consider p being Element of F such that A16: o = -p & p in M
                 by A14a;
             consider sp being set such that A17: p in sp & sp in Y
                  by A16,TARSKI:def 4;
             reconsider so,sp as Preordering of F by AS,A15,A17,H1;
             reconsider o1 = o as Element of F by A14;
             per cases by A15,A17,H5;
             suppose A19: so c= sp;
               o in -sp by A16,A17;
               then o in sp /\ -sp by A19,A15;
               hence o in {0.F} by REALALG1:def 14;
               end;
             suppose sp c= so;
               then o in -so by A16,A17;
               then o in so /\ -so by A15;
               hence o in {0.F} by REALALG1:def 7;
               end;
             end;
           hence thesis by A13,TARSKI:2;
           end;
      SQ F c= M
      proof
        let o be object;
        assume A13: o in SQ F;
        set s = the Element of Y;
        s in Y by H;
        then A15: P c= s by H1,AS;
        SQ F c= P by REALALG1:def 14;
        then o in s by A13,A15;
        hence o in M by H,TARSKI:def 4;
      end;
      hence thesis by A10,A11,A12,REALALG1:def 14;
      end;
  P c= M
    proof
    let o be object;
      assume H0: o in P;
      set s = the Element of Y;
      s in Y by H;
      then P c= s by H1,AS;
      hence o in M by H0,H,TARSKI:def 4;
    end;
  then A4: M in S by A6;
  now let y be set;
      assume A5: y in Y;
      then y c= M by TARSKI:def 4;
      hence [y,M] in R by AS,A4,A5,WELLORD2:def 1;
      end;
  hence ex x being set st x in S &
                         for y being set st y in Y holds [y,x] in R by A4;
  end;
  end;
  hence thesis by ORDERS_1:def 10;
  end;
R is_reflexive_in S & R is_transitive_in S by WELLORD2:19,20;
then R partially_orders S by WELLORD2:21,ORDERS_1:def 8;
then consider x being set such that
M: x is_maximal_in R by A2,A3,ORDERS_1:63;
x in field R by M,ORDERS_1:def 12;
then consider O being Preordering of F such that M1: x = O & P c= O by A2;
M4: O in S by M1;
now let Q be Preordering of F;
  assume M2: O c= Q;
  then P c= Q by M1;
  then M5: Q in S;
  then M3: Q in field R by WELLORD2:def 1;
  [O,Q] in R by M4,M2,M5,WELLORD2:def 1;
  hence O = Q by M3,M1,M,ORDERS_1:def 12;
  end;
hence thesis by M1,defmax;
end;
