
theorem Th106:
  for G1, G2 being _Graph
  for F being non empty PGraphMapping of G1, G2, H2 being Subgraph of rng F
  for H1 being inducedSubgraph of G1,F_V"the_Vertices_of H2,F_E"the_Edges_of H2
  for W1 being Walk of H1 holds W1 is F-defined Walk of G1
proof
  let G1, G2 be _Graph;
  let F be non empty PGraphMapping of G1, G2, H2 be Subgraph of rng F;
  let H1 be inducedSubgraph of G1,F_V"the_Vertices_of H2,F_E"the_Edges_of H2;
  let W1 be Walk of H1;
  A1: the_Vertices_of H1 = F_V"the_Vertices_of H2 &
    the_Edges_of H1 = F_E"the_Edges_of H2
  proof
    set v = the Vertex of H2;
    v in the_Vertices_of H2;
    then v in the_Vertices_of rng F;
    then v in rng F_V by GLIB_010:54;
    then consider x being object such that
      A2: x in dom F_V & F_V.x = v by FUNCT_1:def 3;
    A3: F_V"the_Vertices_of H2 is non empty by A2, FUNCT_1:def 7;
    H2 is Subgraph of G2 by GLIB_000:43;
    then F_E"the_Edges_of H2 c= G1.edgesBetween(F_V"the_Vertices_of H2)
      by Th99;
    hence thesis by A3, GLIB_000:def 37;
  end;
  the_Vertices_of H1 c=dom F_V & the_Edges_of H1 c=dom F_E by A1, RELAT_1:132;
  then A4: W1.vertices() c= dom F_V & W1.edges() c= dom F_E by XBOOLE_1:1;
  reconsider W = W1 as Walk of G1 by GLIB_001:167;
  W.vertices() = W1.vertices() & W.edges() = W1.edges()
    by GLIB_001:98, GLIB_001:110;
  hence thesis by A4, GLIB_010:def 35;
end;
