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

theorem Th21:
  for G being non edgeless _Graph, e being Edge of G
  for V being Subset of the_Vertices_of G
  for H being addVertices of createGraph(e), V
  holds H is Subgraph of G
proof
  let G be non edgeless _Graph, e be Edge of G;
  let V be Subset of the_Vertices_of G;
  let H be addVertices of createGraph(e), V;
  the_Vertices_of H = the_Vertices_of createGraph(e) \/ V by GLIB_006:def 10;
  then A1: the_Vertices_of H c= the_Vertices_of G;
  the_Edges_of H = the_Edges_of createGraph(e) by GLIB_006:def 10;
  then A2: the_Edges_of H c= the_Edges_of G;
  now
    let e0 be set;
    assume e0 in the_Edges_of H;
    then A3: e0 in the_Edges_of createGraph(e) by GLIB_006:def 10;
    thus (the_Source_of H).e0
       = (the_Source_of createGraph(e)).e0 by GLIB_006:def 10
      .= (the_Source_of G).e0 by A3, GLIB_000:def 32;
    thus (the_Target_of H).e0
       = (the_Target_of createGraph(e)).e0 by GLIB_006:def 10
      .= (the_Target_of G).e0 by A3, GLIB_000:def 32;
  end;
  hence thesis by A1, A2, GLIB_000:def 32;
end;
