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

theorem
  for G3 being _Graph, V1, V2 being set, G2 being addVertices of G3, V2,
    G1 being addVertices of G2, V1
  holds G1 is addVertices of G3, V1 \/ V2
proof
  let G3 be _Graph, V1, V2 be set;
  let G2 be addVertices of G3, V2;
  let G1 be addVertices of G2, V1;
  A1: the_Vertices_of G1 = (the_Vertices_of G2) \/ V1 &
    the_Edges_of G1 = the_Edges_of G2 &
    the_Source_of G1 = the_Source_of G2 &
    the_Target_of G1 = the_Target_of G2 by Def10;
  the_Vertices_of G2 = (the_Vertices_of G3) \/ V2 &
    the_Edges_of G2 = the_Edges_of G3 &
    the_Source_of G2 = the_Source_of G3 &
    the_Target_of G2 = the_Target_of G3 by Def10;
  then A3: the_Vertices_of G1 = the_Vertices_of G3 \/ (V1 \/ V2)
    by A1, XBOOLE_1:4;
  A4: the_Edges_of G1 = the_Edges_of G3 by A1, Def10;
  A5: the_Source_of G1 = the_Source_of G3 by A1, Def10;
  A6: the_Target_of G1 = the_Target_of G3 by A1, Def10;
  G1 is Supergraph of G3 by Th66;
  hence thesis by A3, A4, A5, A6, Def10;
end;
