
theorem Th107:
  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 F-defined Walk of G1 st W1 is Walk of H1
  holds F.:W1 is Walk of H2
proof
  let G1, G2 be _Graph;
  let F be non empty 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;
  A1: H2 is Subgraph of G2 by GLIB_000:43;
  let W1 be F-defined Walk of G1;
  assume W1 is Walk of H1;
  then reconsider W = W1 as Walk of H1;
  A2: W.vertices() = W1.vertices() & W.edges() = W1.edges()
    by GLIB_001:98, GLIB_001:110;
  A3: 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
      A4: x in dom F_V & F_V.x = v by FUNCT_1:def 3;
    A5: F_V"the_Vertices_of H2 is non empty by A4, 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 A5, GLIB_000:def 37;
  end;
  A6: (F.:W1).vertices() c= the_Vertices_of H2
  proof
    (F.:W1).vertices() = F_V.:W1.vertices() by GLIB_010:135;
    then A7: (F.:W1).vertices() c= F_V.:the_Vertices_of H1 by A2, RELAT_1:123;
    F_V.:the_Vertices_of H1 c= the_Vertices_of H2 by A3, FUNCT_1:75;
    hence thesis by A7, XBOOLE_1:1;
  end;
  (F.:W1).edges() c= the_Edges_of H2
  proof
    (F.:W1).edges() = F_E.:W1.edges() by GLIB_010:136;
    then A8: (F.:W1).edges() c= F_E.:the_Edges_of H1 by A2, RELAT_1:123;
    F_E.:the_Edges_of H1 c= the_Edges_of H2 by A3, FUNCT_1:75;
    hence thesis by A8, XBOOLE_1:1;
  end;
  hence thesis by A1, A6, GLIB_001:170;
end;
