reserve G for _Graph;
reserve G2 for _Graph, G1 for Supergraph of G2;

theorem Th67:
  for G1, G2 being _Graph st the_Vertices_of G2 c= the_Vertices_of G1 &
    the_Source_of G2 c= the_Source_of G1 &
    the_Target_of G2 c= the_Target_of G1
  holds G1 is Supergraph of G2
proof
  let G1, G2 be _Graph;
  assume that
    A1: the_Vertices_of G2 c= the_Vertices_of G1 and
    A2: the_Source_of G2 c= the_Source_of G1 and
    A3: the_Target_of G2 c= the_Target_of G1;
  A4: dom the_Source_of G2 = the_Edges_of G2 &
    dom the_Target_of G2 = the_Edges_of G2 &
    dom the_Source_of G1 = the_Edges_of G1 &
    dom the_Target_of G1 = the_Edges_of G1 by GLIB_000:4;
  A6: the_Edges_of G2 c= the_Edges_of G1 by A3, A4, RELAT_1:11;
  for e being set st e in the_Edges_of G2 holds
    (the_Source_of G2).e = (the_Source_of G1).e &
    (the_Target_of G2).e = (the_Target_of G1).e by A2, A3, A4, GRFUNC_1:2;
  hence thesis by A1, A6, Def9;
end;
