reserve G, G1, G2 for _Graph, H for Subgraph of G;

theorem
  G is GraphUnion of G.allConnectedSG()
proof
  set G9 = the GraphUnion of G.allConnectedSG();
  G is GraphUnion of G.allSG() by Th35;
  then A1: G9 is Subgraph of G by GLIBPRE1:119;
  now
    A2: for v being Vertex of G holds createGraph(v) in G.allConnectedSG()
      by Th127;
    per cases;
    suppose G is edgeless;
      hence G is Subgraph of G9 by A2, Th24;
    end;
    suppose G is non edgeless;
      then reconsider G0 = G as non edgeless _Graph;
      for e being Edge of G0 holds createGraph(e) in G.allConnectedSG()
        by Th128;
      hence G is Subgraph of G9 by A2, Th25;
    end;
  end;
  hence thesis by A1, GLIB_000:87, GLIB_014:22;
end;
