reserve X,Y,x,y for set;
reserve A for non empty Poset;
reserve a,a1,a2,a3,b,c for Element of A;
reserve S,T for Subset of A;
reserve f for Choice_Function of BOOL(the carrier of A);
reserve fC,fC1,fC2 for Chain of f;

theorem Th37:
  f.(the carrier of A) in fC
proof
  the InternalRel of A well_orders fC & fC <> {} by Def12;
  then consider x being object such that
A1: x in fC and
A2: for y being object st y in fC holds [x,y] in the InternalRel of A
by WELLORD1:5;
  reconsider x as Element of A by A1;
A3: now
    set y = the Element of LowerCone{x} /\ fC;
    assume
A4: LowerCone{x} /\ fC <> {}(A);
    then reconsider a = y as Element of A by Lm1;
    a in LowerCone{x} by A4,XBOOLE_0:def 4;
    then
A5: ex a1 st a = a1 & for a2 st a2 in {x} holds a1 < a2;
    y in fC by A4,XBOOLE_0:def 4;
    then [x,y] in the InternalRel of A by A2;
    then
A6: x <= a;
    x in {x} by TARSKI:def 1;
    hence contradiction by A6,A5,Th6;
  end;
  LowerCone{x} /\ fC = InitSegm(fC,x);
  then f.UpperCone(LowerCone{x} /\ fC) = x by A1,Def12;
  hence thesis by A1,A3,Th14;
end;
