
theorem
  for S1, S2 being Graph-membered set st S1 c= S2 holds
    the_Vertices_of S1 c= the_Vertices_of S2 &
    the_Edges_of S1 c= the_Edges_of S2 &
    the_Source_of S1 c= the_Source_of S2 &
    the_Target_of S1 c= the_Target_of S2
proof
  let S1, S2 be Graph-membered set;
  assume A1: S1 c= S2;
  now
    let x be object;
    assume x in the_Vertices_of S1;
    then consider G being _Graph such that
      A2: G in S1 & x = the_Vertices_of G by GLIB_014:def 14;
    thus x in the_Vertices_of S2 by A1, A2, GLIB_014:def 14;
  end;
  hence the_Vertices_of S1 c= the_Vertices_of S2 by TARSKI:def 3;
  now
    let x be object;
    assume x in the_Edges_of S1;
    then consider G being _Graph such that
      A3: G in S1 & x = the_Edges_of G by GLIB_014:def 15;
    thus x in the_Edges_of S2 by A1, A3, GLIB_014:def 15;
  end;
  hence the_Edges_of S1 c= the_Edges_of S2 by TARSKI:def 3;
  now
    let x be object;
    assume x in the_Source_of S1;
    then consider G being _Graph such that
      A4: G in S1 & x = the_Source_of G by GLIB_014:def 16;
    thus x in the_Source_of S2 by A1, A4, GLIB_014:def 16;
  end;
  hence the_Source_of S1 c= the_Source_of S2 by TARSKI:def 3;
  now
    let x be object;
    assume x in the_Target_of S1;
    then consider G being _Graph such that
      A5: G in S1 & x = the_Target_of G by GLIB_014:def 17;
    thus x in the_Target_of S2 by A1, A5, GLIB_014:def 17;
  end;
  hence the_Target_of S1 c= the_Target_of S2 by TARSKI:def 3;
  thus thesis;
end;
