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
  LowerCone({}(A)) = the carrier of A
proof
  thus LowerCone({}(A)) c= the carrier of A;
  let x be object;
  assume x in the carrier of A;
  then reconsider a = x as Element of A;
  for a2 st a2 in {}(A) holds a < a2;
  hence thesis;
end;
