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 Th36:
  {f.(the carrier of A)} is Chain of f
proof
  set AC = the carrier of A;
  AC in BOOL AC & not {} in BOOL AC by ORDERS_1:1,2;
  then reconsider aa = f.AC as Element of A by ORDERS_1:89;
  reconsider X = {aa} as Chain of A by Th8;
A1: the InternalRel of A is_well_founded_in X
  proof
    reconsider x = aa as set;
    let Y;
    assume that
A2: Y c= X and
A3: Y <> {};
    take x;
    Y = X by A2,A3,ZFMISC_1:33;
    hence x in Y by TARSKI:def 1;
    thus (the InternalRel of A)-Seg(x) /\ Y c= {}
    proof
      let y be object;
      assume
A4:   y in (the InternalRel of A)-Seg(x) /\ Y;
      then y in Y by XBOOLE_0:def 4;
      then
A5:   y = aa by A2,TARSKI:def 1;
      y in (the InternalRel of A)-Seg(x) by A4,XBOOLE_0:def 4;
      hence thesis by A5,WELLORD1:1;
    end;
    thus thesis;
  end;
A6: for a st a in X holds f.UpperCone(InitSegm(X,a)) = a
  proof
    let a;
    assume a in X;
    then
A7: a = aa by TARSKI:def 1;
    LowerCone{a} /\ X c= {}(A)
    proof
      let x be object;
      assume
A8:   x in LowerCone{a} /\ X;
      then x in LowerCone{a} by XBOOLE_0:def 4;
      then
A9:   ex a1 st x = a1 & for a2 st a2 in {a} holds a1 < a2;
      x in X by A8,XBOOLE_0:def 4;
      hence thesis by A7,A9;
    end;
    then LowerCone{a} /\ X = {}(A);
    hence thesis by A7,Th14;
  end;
  the InternalRel of A is_strongly_connected_in X by Def7;
  then
A10: the InternalRel of A is_connected_in X;
  the InternalRel of A is_antisymmetric_in the carrier of A by Def4;
  then
A11: the InternalRel of A is_antisymmetric_in X;
  the InternalRel of A is_transitive_in the carrier of A by Def3;
  then
A12: the InternalRel of A is_transitive_in X;
  the InternalRel of A is_reflexive_in the carrier of A by Def2;
  then the InternalRel of A is_reflexive_in X;
  then the InternalRel of A well_orders X by A12,A11,A10,A1;
  hence thesis by A6,Def12;
end;
