reserve D for non empty set,
  f for FinSequence of D,
  p, p1, p2, p3, q for Element of D,
  i, j, k, l, n for Nat;

theorem
  1 <= i & i <= len f implies for k st 0 < k & k <= len f - i holds
  Replace(f, i, p).(i + k) = (f/^i).k
proof
  assume that
A1: 1 <= i and
A2: i <= len f;
  i - 1 < i by XREAL_1:146;
  then
A3: i -' 1 < i by A1,XREAL_1:233;
A4: len ((f|(i-'1))^<*p*>) = len (f|(i-'1)) + len <*p*> by FINSEQ_1:22
    .= (i -' 1) + len <*p*> by A2,A3,FINSEQ_1:59,XXREAL_0:2
    .= i -' 1 + 1 by FINSEQ_1:39
    .= i by A1,XREAL_1:235;
  let k;
  assume that
A5: 0 < k and
A6: k <= len f - i;
A7: 0 + 1 <= k by A5,INT_1:7;
  len f = len Replace(f, i, p) by FUNCT_7:97
    .= len ((f|(i-'1))^<*p*>^(f/^i)) by A1,A2,Def1
    .= i + len (f/^i) by A4,FINSEQ_1:22;
  then
A8: k in dom (f/^i) by A6,A7,FINSEQ_3:25;
  Replace(f, i, p).(i + k) = ((f|(i-'1))^<*p*>^(f/^i)).(len ((f|(i-'1))^
  <*p*>) + k) by A1,A2,A4,Def1;
  hence thesis by A8,FINSEQ_1:def 7;
end;
