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

theorem Th25:
  for G being non edgeless _Graph, S being GraphUnionSet
  for G9 being GraphUnion of S
  st (for v being Vertex of G holds createGraph(v) in S) &
    (for e being Edge of G holds createGraph(e) in S)
  holds G is Subgraph of G9
proof
  let G be non edgeless _Graph, S be GraphUnionSet, G9 be GraphUnion of S;
  assume that
    A1: for v being Vertex of G holds createGraph(v) in S and
    A2: for e being Edge of G holds createGraph(e) in S;
  A3: now
    let v be Vertex of G;
    reconsider H9 = createGraph(v) as Element of S by A1;
    take H9;
    thus v in the_Vertices_of H9 by ZFMISC_1:31;
  end;
  now
    let e be Edge of G;
    reconsider H9 = createGraph(e) as Element of S by A2;
    take H9;
    thus createGraph(e) is Subgraph of H9 by GLIB_000:40;
  end;
  hence thesis by A3, Th23;
end;
