
theorem Th50:
  for G being _Graph, W1, W2, W3 being Walk of G, u,v being object holds
    W1 is_Walk_from u,v iff W1.replaceWith(W2, W3) is_Walk_from u,v
proof
  let G be _Graph;
  let W1, W2, W3 be Walk of G;
  let u,v be object;
  W1 is_Walk_from u,v iff W1.first() = u & W1.last() = v by GLIB_001:def 23;
  then W1 is_Walk_from u,v iff
    u = W1.replaceWith(W2, W3).first() &
    v = W1.replaceWith(W2, W3).last() by Th39;
  hence thesis by GLIB_001:def 23;
end;
