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 Th38:
  a in fC & b = f.(the carrier of A) implies b <= a
proof
  assume that
A1: a in fC and
A2: b = f.(the carrier of A);
  the InternalRel of A well_orders fC & fC <> {} by Def12;
  then consider x being object such that
A3: x in fC and
A4: 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 A3;
A5: now
    set y = the Element of LowerCone{x} /\ fC;
    assume
A6: LowerCone{x} /\ fC <> {}(A);
    then reconsider a = y as Element of A by Lm1;
    a in LowerCone{x} by A6,XBOOLE_0:def 4;
    then
A7: ex a1 st a = a1 & for a2 st a2 in {x} holds a1 < a2;
    y in fC by A6,XBOOLE_0:def 4;
    then [x,y] in the InternalRel of A by A4;
    then
A8: x <= a;
    x in {x} by TARSKI:def 1;
    hence contradiction by A8,A7,Th6;
  end;
  LowerCone{x} /\ fC = InitSegm(fC,x);
  then f.UpperCone(LowerCone{x} /\ fC) = x by A3,Def12;
  then
A9: f.(the carrier of A) = x by A5,Th14;
  [x,a] in the InternalRel of A by A1,A4;
  hence thesis by A2,A9;
end;
