
theorem
  for G1, G2 being VGraph, H being VSubgraph of G1,
    F being PGraphMapping of G1, G2
  st F is vlabel-preserving holds F | H is vlabel-preserving
proof
  let G1, G2 be VGraph, H being VSubgraph of G1;
  let F be PGraphMapping of G1, G2;
  assume A1: F is vlabel-preserving;
  the_VLabel_of G2 * (F|H)_V
     = (the_VLabel_of G2 * F_V) | the_Vertices_of H by RELAT_1:83
    .= (the_VLabel_of G1) | (
      dom F_V /\ (the_Vertices_of H /\ the_Vertices_of H))  by A1, RELAT_1:71
    .= (the_VLabel_of G1) | (
      the_Vertices_of H /\ (dom F_V /\ the_Vertices_of H)) by XBOOLE_1:16
    .= (the_VLabel_of G1) | (the_Vertices_of H /\ dom (F|H)_V) by Th59
    .= ((the_VLabel_of G1) | the_Vertices_of H) | dom (F|H)_V by RELAT_1:71
    .= (the_VLabel_of H) | dom (F|H)_V by GLIB_003:def 12;
  hence thesis;
end;
