reserve x,y,z for set;
reserve f,f1,f2,f3 for FinSequence,
  p,p1,p2,p3 for set,
  i,k for Nat;
reserve D for non empty set,
  p,p1,p2,p3 for Element of D,
  f,f1,f2 for FinSequence of D;
reserve D for non empty set;
reserve p, q for FinSequence,
  X, Y, x, y for set,
  D for non empty set,
  i, j, k, l, m, n, r for Nat;
reserve a, a1, a2 for TwoValued Alternating FinSequence;
reserve fs, fs1, fs2 for FinSequence of X,
  fss, fss2 for Subset of fs;
reserve F, F1 for FinSequence of INT,
  k, m, n, ma for Nat;

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;
