
theorem Th132:
  for G1, G2 being _Graph, F being non empty PGraphMapping of G1, G2,
    W1 being F-defined Walk of G1, v, w being object
  holds W1 is_Walk_from v,w implies F.:W1 is_Walk_from F_V.v, F_V.w
proof
  let G1, G2 be _Graph;
  let F be non empty PGraphMapping of G1, G2;
  let W1 be F-defined Walk of G1, v, w be object;
  assume W1 is_Walk_from v,w;
  then W1.first() = v & W1.last() = w by GLIB_001:def 23;
  then W1.1 = v & W1.len W1 = w by GLIB_001:def 6, GLIB_001:def 7;
  then F_V.v = (F.:W1).1 & F_V.w = (F.:W1).len W1 by Th129
, ABIAN:12,POLYFORM:4;
  then F_V.v = (F.:W1).first() & F_V.w = (F.:W1).len (F.:W1)
    by GLIB_001:def 6, Th125;
  then F_V.v = (F.:W1).first() & F_V.w = (F.:W1).last() by GLIB_001:def 7;
  hence F.:W1 is_Walk_from F_V.v, F_V.w by GLIB_001:def 23;
end;
