
theorem Th125:
  for G1, G2 being _Graph, F being non empty PGraphMapping of G1, G2,
    W1 being F-defined Walk of G1
  holds W1.length() = (F.:W1).length() & len W1 = len (F.:W1)
proof
  let G1, G2 be _Graph, F be non empty PGraphMapping of G1, G2;
  let W1 be F-defined Walk of G1;
  W1.edges() = rng W1.edgeSeq() by GLIB_001:def 17;
  then A1: rng W1.edgeSeq() c= dom F_E by Def35;
  then reconsider p = F_E * W1.edgeSeq() as FinSequence by FINSEQ_1:16;
  thus A2: W1.length() = len W1.edgeSeq() by GLIB_001:def 18
    .= len p by A1, FINSEQ_2:29
    .= len (F.:W1).edgeSeq() by Def37
    .= (F.:W1).length() by GLIB_001:def 18;
  thus len W1 = 2*len W1.edgeSeq() + 1 by GLIB_001:def 15
    .= 2*W1.length() + 1 by GLIB_001:def 18
    .= 2*len (F.:W1).edgeSeq() + 1 by A2, GLIB_001:def 18
    .= len (F.:W1) by GLIB_001:def 15;
end;
