
theorem Th22:
  for S being GraphUnionSet, G being GraphUnion of S, G9 being _Graph
  holds G9 is GraphUnion of S iff G == G9
proof
  let S be GraphUnionSet, G be GraphUnion of S, G9 be _Graph;
  A1: the_Vertices_of G = union the_Vertices_of S &
    the_Edges_of G = union the_Edges_of S &
    the_Source_of G = union the_Source_of S &
    the_Target_of G = union the_Target_of S by Def25;
  hereby
    assume G9 is GraphUnion of S;
    then the_Vertices_of G9 = union the_Vertices_of S &
      the_Edges_of G9 = union the_Edges_of S &
      the_Source_of G9 = union the_Source_of S &
      the_Target_of G9 = union the_Target_of S by Def25;
    hence G == G9 by A1, GLIB_000:def 34;
  end;
  assume G == G9;
  then the_Vertices_of G = the_Vertices_of G9 &
    the_Edges_of G = the_Edges_of G9 &
    the_Source_of G = the_Source_of G9 &
    the_Target_of G = the_Target_of G9 by GLIB_000:def 34;
  hence thesis by A1, Def25;
end;
