
theorem
  for G1, G2 being _Graph, F being non empty PGraphMapping of G1, G2
  for V being non empty Subset of the_Vertices_of dom F
  for H being inducedSubgraph of G1, V st F is continuous
  holds rng(F | H) is inducedSubgraph of G2, F_V.:V
proof
  let G1, G2 be _Graph, F be non empty PGraphMapping of G1, G2;
  let V be non empty Subset of the_Vertices_of dom F;
  A1: V is non empty Subset of the_Vertices_of G1 by XBOOLE_1:1;
  let H be inducedSubgraph of G1, V;
  assume A2: F is continuous;
  set v = the Vertex of H;
  v in the_Vertices_of H;
  then v in V by A1, GLIB_000:def 37;
  then v in the_Vertices_of H /\ the_Vertices_of dom F by XBOOLE_0:def 4;
  then v in the_Vertices_of H /\ dom F_V by GLIB_010:54;
  then v in dom((F|H)_V) by RELAT_1:61;
  then A3: F | H is non empty;
  then A4: the_Vertices_of rng(F|H) = rng((F|H)_V) by GLIB_010:54
    .= F_V.:the_Vertices_of H by RELAT_1:115
    .= F_V.:V by A1, GLIB_000:def 37;
  the_Edges_of rng(F|H) = rng((F|H)_E) by A3, GLIB_010:54
    .= F_E.:the_Edges_of H by RELAT_1:115
    .= F_E.:G1.edgesBetween(V) by A1, GLIB_000:def 37
    .= G2.edgesBetween(F_V.:V) by A2, Th104;
  hence thesis by A4, GLIB_000:def 37;
end;
