reserve G for Graph,
  k, m, n for Nat;
reserve G for non void Graph;

theorem
  for f being FinSequence holds len (m,n)-cut f <= len f
proof
  let f be FinSequence;
  set lmnf = len (m,n)-cut f;
  set lf = len f;
  per cases;
  suppose
A1: 1<=m & m<=n & n<=len f;
    then lmnf +m = n+1 by FINSEQ_6:def 4;
    then n+(lmnf +m) <= n+1+lf by A1,XREAL_1:6;
    then n+(lmnf +m) <= n+(1+lf);
    then lmnf +m <= 1+lf by XREAL_1:6;
    then (lmnf +m)+1 <= m+(1+lf) by A1,XREAL_1:7;
    then lmnf +(m+1) <= m+1+lf;
    hence thesis by XREAL_1:6;
  end;
  suppose
    not (1<=m & m<=n & n<=len f);
    then (m,n)-cut f is empty by FINSEQ_6:def 4;
    hence thesis;
  end;
end;
