
theorem Th138:
  for G1, G2 being _Graph,
    F being non empty one-to-one PGraphMapping of G1, G2,
    W1 being F-defined Walk of G1
  holds
    (W1 is trivial iff F.:W1 is trivial) &
    (W1 is closed iff F.:W1 is closed) &
    (W1 is Trail-like iff F.:W1 is Trail-like) &
    (W1 is Path-like iff F.:W1 is Path-like) &
    (W1 is Circuit-like iff F.:W1 is Circuit-like) &
    (W1 is Cycle-like iff F.:W1 is Cycle-like)
proof
  let G1, G2 be _Graph;
  let F be non empty one-to-one PGraphMapping of G1, G2;
  let W1 be F-defined Walk of G1;
  W1 = F"(F.:W1) by Th123;
  hence A1:
    (W1 is trivial iff F.:W1 is trivial) &
    (W1 is closed iff F.:W1 is closed) &
    (W1 is Trail-like iff F.:W1 is Trail-like) &
    (W1 is Path-like iff F.:W1 is Path-like) by Th137;
  thus W1 is Circuit-like iff F.:W1 is Circuit-like by A1;
  thus W1 is Cycle-like iff F.:W1 is Cycle-like by A1;
end;
