
theorem Th67:
  for S being non empty connected Graph-membered set
  holds the set of all H.componentSet() where H is Element of S
    = SmallestPartition the_Vertices_of S
proof
  let S be non empty connected Graph-membered set;
  set M1 = the set of all H.componentSet() where H is Element of S;
  set M2 = the set of all {X} where X is Element of the_Vertices_of S;
  now
    let x be object;
    hereby
      assume x in M1;
      then consider H being Element of S such that
        A1: x = H.componentSet();
      A2: x = {the_Vertices_of H} by A1, GLIB_002:25;
      the_Vertices_of H in the_Vertices_of S by GLIB_014:def 14;
      hence x in M2 by A2;
    end;
    assume x in M2;
    then consider X being Element of the_Vertices_of S such that
      A3: x = {X};
    consider H being _Graph such that
      A4: H in S & X = the_Vertices_of H by GLIB_014:def 14;
    reconsider H as Element of S by A4;
    x = H.componentSet() by A3, A4, GLIB_002:25;
    hence x in M1;
  end;
  then M1 = M2 by TARSKI:2;
  hence thesis by EQREL_1:37;
end;
