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;

theorem Th11:
  for A being reflexive RelStr, a1,a2 being Element of A holds (ex
  C being Chain of A st a1 in C & a2 in C) iff a1 <= a2 or a2 <= a1
proof
  let A be reflexive RelStr;
  let a1,a2 be Element of A;
  thus (ex C being Chain of A st a1 in C & a2 in C) implies a1 <= a2 or a2 <=
  a1
  proof
    given C being Chain of A such that
A1: a1 in C & a2 in C;
    the InternalRel of A is_strongly_connected_in C by Def7;
    then
    [a1,a2] in the InternalRel of A or [a2,a1] in the InternalRel of A by A1;
    hence thesis;
  end;
  assume
A2: a1 <= a2 or a2 <= a1;
  then
  [a1,a2] in the InternalRel of A or [a2,a1] in the InternalRel of A;
  then reconsider B = A as non empty reflexive RelStr;
  reconsider b1 = a1, b2 = a2 as Element of B;
  reconsider Y = {b1,b2} as Chain of A by A2,Th9;
  take Y;
  thus thesis by TARSKI:def 2;
end;
