theorem
  for f being FinSequence of INT,m,n be Nat st m >= n
  holds f is_non_decreasing_on m,n
proof
  let f be FinSequence of INT,m,n be Nat;
  assume
A1: m>=n;
    let i, j be Nat such that
A2: m <= i & i <= j and
A3: j <= n;
A4: m <= j by A2,XXREAL_0:2;
  per cases by A1,XXREAL_0:1;
  suppose m=n;
    then j=m by A3,A4,XXREAL_0:1;
    hence f.i <= f.j by A2,XXREAL_0:1;
  end;
  suppose m>n;
    hence thesis by A3,A4,XXREAL_0:2;
  end;
end;
