
theorem
  for G1, G2 being _Graph, F being non empty PGraphMapping of G1, G2
  for W1 being F-defined Walk of G1
  holds (F.:W1).vertices() = F_V.:W1.vertices()
proof
  let G1, G2 be _Graph, F be non empty PGraphMapping of G1, G2;
  let W1 be F-defined Walk of G1;
  A1: (F.:W1).vertices() = rng((F.:W1).vertexSeq()) by GLIB_001:def 16
    .= rng(F_V * W1.vertexSeq()) by Def37;
  for y being object holds y in rng(F_V * W1.vertexSeq())
    iff y in F_V.:W1.vertices()
  proof
    let y be object;
    hereby
      assume y in rng(F_V * W1.vertexSeq());
      then consider x being object such that
        A2: x in dom(F_V * W1.vertexSeq()) & (F_V * W1.vertexSeq()).x = y
        by FUNCT_1:def 3;
      set v = W1.vertexSeq().x;
      x in dom W1.vertexSeq() by A2, FUNCT_1:11;
      then v in rng W1.vertexSeq() by FUNCT_1:3;
      then A3: v in W1.vertices() by GLIB_001:def 16;
      A4: v in dom F_V by A2, FUNCT_1:11;
      F_V.v = y by A2, FUNCT_1:12;
      hence y in F_V.:W1.vertices() by A3, A4, FUNCT_1:def 6;
    end;
    assume y in F_V.:W1.vertices();
    then consider v being object such that
      A5: v in dom F_V & v in W1.vertices() & F_V.v = y by FUNCT_1:def 6;
    v in rng W1.vertexSeq() by A5, GLIB_001:def 16;
    then consider x being object such that
      A6: x in dom W1.vertexSeq() & W1.vertexSeq().x = v by FUNCT_1:def 3;
    A7: (F_V * W1.vertexSeq()).x = y by A5, A6, FUNCT_1:13;
    x in dom(F_V * W1.vertexSeq()) by A5, A6, FUNCT_1:11;
    hence y in rng(F_V * W1.vertexSeq()) by A7, FUNCT_1:3;
  end;
  hence thesis by A1, TARSKI:2;
end;
