
theorem
  for G, H being _Graph holds G is GraphUnion of {H} iff G == H
proof
  let G, H be _Graph;
  hereby
    assume G is GraphUnion of {H};
    then the_Vertices_of G = union the_Vertices_of {H} &
      the_Edges_of G = union the_Edges_of {H} &
      the_Source_of G = union the_Source_of {H} &
      the_Target_of G = union the_Target_of {H} by Def25;
    then the_Vertices_of G = union {the_Vertices_of H} &
      the_Edges_of G = union {the_Edges_of H} &
      the_Source_of G = union {the_Source_of H} &
      the_Target_of G = union {the_Target_of H} by Th5;
    then the_Vertices_of G = the_Vertices_of H &
      the_Edges_of G = the_Edges_of H &
      the_Source_of G = the_Source_of H &
      the_Target_of G = the_Target_of H by ZFMISC_1:25;
    hence G == H by GLIB_000:def 34;
  end;
  assume G == H;
  then the_Vertices_of G = the_Vertices_of H &
    the_Edges_of G = the_Edges_of H &
    the_Source_of G = the_Source_of H &
    the_Target_of G = the_Target_of H by GLIB_000:def 34;
  then the_Vertices_of G = union {the_Vertices_of H} &
    the_Edges_of G = union {the_Edges_of H} &
    the_Source_of G = union {the_Source_of H} &
    the_Target_of G = union {the_Target_of H} by ZFMISC_1:25;
  then the_Vertices_of G = union the_Vertices_of {H} &
    the_Edges_of G = union the_Edges_of {H} &
    the_Source_of G = union the_Source_of {H} &
    the_Target_of G = union the_Target_of {H} by Th5;
  hence G is GraphUnion of {H} by Def25;
end;
