
theorem Th131:
  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 v in dom F_V & w in dom F_V
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 v in W1.vertices() & w in W1.vertices() by GLIB_001:88;
  hence v in dom F_V & w in dom F_V by Def35, TARSKI:def 3;
end;
