
theorem Th21:
  for S being GraphUnionSet, G being GraphUnion of S, H being Element of S
  holds H is Subgraph of G
proof
  let S be GraphUnionSet, G be GraphUnion of S, H be Element of S;
  the_Vertices_of H in the_Vertices_of S;
  then the_Vertices_of H c= union the_Vertices_of S by ZFMISC_1:74;
  then A1: the_Vertices_of H c= the_Vertices_of G by Def25;
  the_Source_of H in the_Source_of S;
  then the_Source_of H c= union the_Source_of S by ZFMISC_1:74;
  then A2: the_Source_of H c= the_Source_of G by Def25;
  the_Target_of H in the_Target_of S;
  then the_Target_of H c= union the_Target_of S by ZFMISC_1:74;
  then the_Target_of H c= the_Target_of G by Def25;
  then G is Supergraph of H by A1, A2, GLIB_006:63;
  hence thesis by GLIB_006:57;
end;
