
theorem Th109:
  for G1, G2 being _Graph, F being non empty one-to-one PGraphMapping of G1,G2
  for 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 W2 being F-valued Walk of G2 st W2 is Walk of H2
  holds F"W2 is Walk of H1
proof
  let G1, G2 be _Graph, F be non empty one-to-one PGraphMapping of G1, G2;
  let 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 W2 be F-valued Walk of G2;
  assume W2 is Walk of H2;
  then reconsider W = W2 as Walk of H2;
  A1: W.vertices() = W2.vertices() & W.edges() = W2.edges()
    by GLIB_001:98, GLIB_001:110;
  A2: 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
      A3: x in dom F_V & F_V.x = v by FUNCT_1:def 3;
    A4: F_V"the_Vertices_of H2 is non empty by A3, 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 A4, GLIB_000:def 37;
  end;
  A5: (F"W2).vertices() c= the_Vertices_of H1
  proof
    (F"W2).vertices() = F_V"W2.vertices() by Th95;
    hence thesis by A1, A2, RELAT_1:143;
  end;
  (F"W2).edges() c= the_Edges_of H1
  proof
    (F"W2).edges() = F_E"W2.edges() by Th96;
    hence thesis by A1, A2, RELAT_1:143;
  end;
  hence thesis by A5, GLIB_001:170;
end;
