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, G1 being addVertices of G3, V1 \/ V2
  holds ex G2 being addVertices of G3, V2 st G1 is addVertices of G2, V1
proof
  let G3 be _Graph, V1, V2 be set;
  let G1 be addVertices of G3, V1 \/ V2;
  set G2 = the addVertices of G3, V2;
  take G2;
  A1: 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;
  A2: the_Vertices_of G1 = (the_Vertices_of G3) \/ (V1 \/ V2) &
    the_Edges_of G1 = the_Edges_of G3 &
    the_Source_of G1 = the_Source_of G3 &
    the_Target_of G1 = the_Target_of G3 by Def10;
  (the_Vertices_of G3) \/ V2 c= (the_Vertices_of G3) \/ (V1 \/ V2)
    by XBOOLE_1:7, XBOOLE_1:9;
  then A4: the_Vertices_of G2 c= the_Vertices_of G1 by A1, Def10;
  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 A1, A2;
  then A5: G1 is Supergraph of G2 by A1, A2, A4, Def9;
  the_Vertices_of G1 = (the_Vertices_of G3 \/ V2) \/ V1 by A2, XBOOLE_1:4
    .= the_Vertices_of G2 \/ V1 by Def10;
  hence thesis by A1, A2, A5, Def10;
end;
