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
  a = f.(the carrier of A) implies InitSegm(fC,a) = {}
proof
  set x = the Element of LowerCone{a} /\ fC;
  assume
A1: a = f.(the carrier of A);
  then
A2: a in fC by Th37;
  assume
A3: InitSegm(fC,a) <> {};
  then reconsider b = x as Element of A by Lm1;
  x in LowerCone{a} by A3,XBOOLE_0:def 4;
  then
A4: ex a1 st a1 = b & for a2 st a2 in {a} holds a1 < a2;
  a in {a} by TARSKI:def 1;
  then
A5: b < a by A4;
A6: x in fC by A3,XBOOLE_0:def 4;
  then a <= b by A1,Th38;
  hence contradiction by A2,A6,A5,Th12;
end;
