
theorem Th121:
  for S1, S2 being GraphUnionSet
  for G1 being (GraphUnion of S1), G2 being GraphUnion of S2
  st (for H2 being Element of S2 ex H1 being Element of S1
    st H2 is Subgraph of H1)
  holds G2 is Subgraph of G1
proof
  let S1, S2 be GraphUnionSet;
  let G1 be (GraphUnion of S1), G2 be GraphUnion of S2;
  assume A1: for H2 being Element of S2 ex H1 being Element of S1
    st H2 is Subgraph of H1;
  now
    now
      let x be object;
      assume x in the_Vertices_of G2;
      then x in union the_Vertices_of S2 by GLIB_014:def 25;
      then consider V being set such that
        A2: x in V & V in the_Vertices_of S2 by TARSKI:def 4;
      consider H2 being _Graph such that
        A3: H2 in S2 & V = the_Vertices_of H2 by A2, GLIB_014:def 14;
      consider H1 being Element of S1 such that
        A4: H2 is Subgraph of H1 by A1, A3;
      the_Vertices_of H2 c= the_Vertices_of H1 by A4, GLIB_000:def 32;
      then A5: x in the_Vertices_of H1 by A2, A3;
      the_Vertices_of H1 in the_Vertices_of S1 by GLIB_014:def 14;
      then x in union the_Vertices_of S1 by A5, TARSKI:def 4;
      hence x in the_Vertices_of G1 by GLIB_014:def 25;
    end;
    hence the_Vertices_of G2 c= the_Vertices_of G1 by TARSKI:def 3;
    now
      let x be object;
      assume x in the_Edges_of G2;
      then x in union the_Edges_of S2 by GLIB_014:def 25;
      then consider E being set such that
        A6: x in E & E in the_Edges_of S2 by TARSKI:def 4;
      consider H2 being _Graph such that
        A7: H2 in S2 & E = the_Edges_of H2 by A6, GLIB_014:def 15;
      consider H1 being Element of S1 such that
        A8: H2 is Subgraph of H1 by A1, A7;
      the_Edges_of H2 c= the_Edges_of H1 by A8, GLIB_000:def 32;
      then A9: x in the_Edges_of H1 by A6, A7;
      the_Edges_of H1 in the_Edges_of S1 by GLIB_014:def 15;
      then x in union the_Edges_of S1 by A9, TARSKI:def 4;
      hence x in the_Edges_of G1 by GLIB_014:def 25;
    end;
    hence the_Edges_of G2 c= the_Edges_of G1 by TARSKI:def 3;
    let e be set;
    set v = (the_Source_of G2).e, w = (the_Target_of G2).e;
    assume e in the_Edges_of G2;
    then consider H2 being Element of S2 such that
      A10: e DJoins v,w,H2 by Th119, GLIB_000:def 14;
    consider H1 being Element of S1 such that
      A11: H2 is Subgraph of H1 by A1;
    H1 is Subgraph of G1 by GLIB_014:21;
    then H2 is Subgraph of G1 by A11, GLIB_000:43;
    then e DJoins v,w,G1 by A10, GLIB_000:72;
    hence (the_Source_of G2).e = (the_Source_of G1).e &
      (the_Target_of G2).e = (the_Target_of G1).e by GLIB_000:def 14;
  end;
  hence thesis by GLIB_000:def 32;
end;
