reserve X for set,
        D for a_partition of X,
        TG for non empty TopologicalGroup;
reserve A for Subset of X;

theorem
  for SF being upper Subset-Family of X,
      S being Element of SF holds meet SF c= S
  proof
    let SF be upper Subset-Family of X, S be Element of SF;
    let x be object;
    assume
A1: x in meet SF;
    per cases;
    suppose SF is empty;
      hence thesis by A1,SETFAM_1:def 1;
    end;
    suppose SF is non empty;
      hence thesis by A1,SETFAM_1:def 1;
    end;
  end;
