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 Th23:
  a < c iff a in LowerCone{c}
proof
  thus a < c implies a in LowerCone{c}
  proof
    assume a < c;
    then for b holds b in {c} implies a < b by TARSKI:def 1;
    hence thesis;
  end;
  assume a in LowerCone{c};
  then
A1: ex a1 st a1 = a & for a2 st a2 in {c} holds a1 < a2;
  c in {c} by TARSKI:def 1;
  hence thesis by A1;
end;
