
theorem Th108:
  for G1, G2 being _Graph, F being non empty PGraphMapping of G1, G2
  for H being Subgraph of rng F, W being Walk of H
  holds W is F-valued Walk of G2
proof
  let G1, G2 be _Graph, F be non empty PGraphMapping of G1, G2;
  let H be Subgraph of rng F, W be Walk of H;
  H is Subgraph of G2 by GLIB_000:43;
  then reconsider W9 = W as Walk of G2 by GLIB_001:167;
  the_Vertices_of H c= the_Vertices_of rng F &
    the_Edges_of H c= the_Edges_of rng F;
  then W.vertices() c= the_Vertices_of rng F &
    W.edges() c= the_Edges_of rng F by XBOOLE_1:1;
  then W.vertices() c= rng F_V & W.edges() c= rng F_E by GLIB_010:54;
  then W9.vertices() c= rng F_V & W9.edges() c= rng F_E
    by GLIB_001:98, GLIB_001:110;
  hence thesis by GLIB_010:def 36;
end;
