
theorem Th10:
  for G being _Graph, W be Walk of G, m, n being Nat holds len W
  .cut(m,n) <= len W
proof
  let G be _Graph, W be Walk of G, m, n be Nat;
  reconsider m9 = m, n9 = n as Element of NAT by ORDINAL1:def 12;
  per cases;
  suppose
    m is odd & n is odd & m <= n & n <= len W;
    then W.cut(m,n) = (m,n)-cut W by GLIB_001:def 11;
    then len W.cut(m9,n9) <= len W by MSSCYC_1:8;
    hence thesis;
  end;
  suppose
    not (m is odd & n is odd & m <= n & n <= len W);
    hence thesis by GLIB_001:def 11;
  end;
end;
