
theorem Th136:
  for G1, G2 being _Graph, F being non empty PGraphMapping of G1, G2
  for W1 being F-defined Walk of G1
  holds (F.:W1).edges() = F_E.:W1.edges()
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).edges() = rng((F.:W1).edgeSeq()) by GLIB_001:def 17
    .= rng(F_E * W1.edgeSeq()) by Def37;
  for y being object holds y in rng(F_E * W1.edgeSeq())
    iff y in F_E.:W1.edges()
  proof
    let y be object;
    hereby
      assume y in rng(F_E * W1.edgeSeq());
      then consider x being object such that
        A2: x in dom(F_E * W1.edgeSeq()) & (F_E * W1.edgeSeq()).x = y
        by FUNCT_1:def 3;
      set v = W1.edgeSeq().x;
      x in dom W1.edgeSeq() by A2, FUNCT_1:11;
      then v in rng W1.edgeSeq() by FUNCT_1:3;
      then A3: v in W1.edges() by GLIB_001:def 17;
      A4: v in dom F_E by A2, FUNCT_1:11;
      F_E.v = y by A2, FUNCT_1:12;
      hence y in F_E.:W1.edges() by A3, A4, FUNCT_1:def 6;
    end;
    assume y in F_E.:W1.edges();
    then consider v being object such that
      A5: v in dom F_E & v in W1.edges() & F_E.v = y by FUNCT_1:def 6;
    v in rng W1.edgeSeq() by A5, GLIB_001:def 17;
    then consider x being object such that
      A6: x in dom W1.edgeSeq() & W1.edgeSeq().x = v by FUNCT_1:def 3;
    A7: (F_E * W1.edgeSeq()).x = y by A5, A6, FUNCT_1:13;
    x in dom(F_E * W1.edgeSeq()) by A5, A6, FUNCT_1:11;
    hence y in rng(F_E * W1.edgeSeq()) by A7, FUNCT_1:3;
  end;
  hence thesis by A1, TARSKI:2;
end;
