
theorem Th23:
  for G being _Graph holds G.componentSet() is a_partition of the_Vertices_of G
proof
  let G be _Graph;
  set V = the_Vertices_of G;
  A1: union G.componentSet() = V by GLIB_002:24;
  for A being Subset of V st A in G.componentSet() holds A <> {} &
    for B being Subset of V st B in G.componentSet() holds A = B or A misses B
  proof
    let A be Subset of V;
    assume A in G.componentSet();
    then consider v being Vertex of G such that
      A2: A = G.reachableFrom(v) by GLIB_002:def 8;
    thus A <> {} by A2;
    let B be Subset of V;
    assume B in G.componentSet();
    then consider w being Vertex of G such that
      A3: B = G.reachableFrom(w) by GLIB_002:def 8;
    per cases;
    suppose v in G.reachableFrom(w);
      hence thesis by A2, A3, GLIB_002:12;
    end;
    suppose not v in G.reachableFrom(w);
      hence thesis by A2, A3, Th22;
    end;
  end;
  hence thesis by A1, EQREL_1:def 4;
end;
