
theorem
  for G1, G2 being _Graph,
    F being non empty one-to-one PGraphMapping of G1, G2,
    W1 being F-defined Walk of G1, v, w being object
  holds W1 is_Walk_from v,w iff v in dom F_V & w in dom F_V
    & F.:W1 is_Walk_from F_V.v, F_V.w
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, v, w be object;
  thus W1 is_Walk_from v,w implies v in dom F_V & w in dom F_V &
    F.:W1 is_Walk_from F_V.v, F_V.w by Th131, Th132;
  assume that
    A1: v in dom F_V & w in dom F_V and
    A2: F.:W1 is_Walk_from F_V.v, F_V.w;
  F"(F.:W1) is_Walk_from F"_V.(F_V.v), F"_V.(F_V.w) by A2, Th132;
  then W1 is_Walk_from F_V".(F_V.v), F_V".(F_V.w) by Th123;
  then W1 is_Walk_from v, F_V".(F_V.w) by A1, FUNCT_1:34;
  hence thesis by A1, FUNCT_1:34;
end;
