
theorem Th49:
  for G1, G2 being _Graph, W1 being Walk of G1, W2 being Walk of G2
  st G1 == G2 & W1 = W2 & W1 is minlength holds W2 is minlength
proof
  let G1, G2 be _Graph, W1 be Walk of G1, W2 be Walk of G2;
  assume A1: G1 == G2 & W1 = W2 & W1 is minlength;
  now
    let W4 be Walk of G2;
    assume A2: W4 is_Walk_from W2.first(),W2.last();
    reconsider W3 = W4 as Walk of G1 by A1, GLIB_001:179;
    W3 is_Walk_from W2.first(),W2.last() by A2, GLIB_001:19;
    then W3 is_Walk_from W1.first(),W2.last() by A1;
    then W3 is_Walk_from W1.first(),W1.last() by A1;
    hence len W4 >= len W2 by A1, CHORD:def 2;
  end;
  hence thesis by CHORD:def 2;
end;
