
theorem
  for k,n be Nat holds
    (Rascal (k + n)).(n+1) * (Rascal (k + n + 2)).(n+2) =
      (Rascal (k + n + 1)).(n+1) * (Rascal (k + n + 1)).(n + 2) + 1
  proof
    let k,n be Nat;
    A0: dom(Rascal (k+n)) = Seg (k + n + 1) &
      dom (Rascal (k + n + 1)) = Seg(k + n + 1 + 1) &
    dom (Rascal (k+n+2)) = Seg (k + n + 2 + 1) by Def1;
    1 + 0 <= 1 + n & 0 + (n + 1) <= k + (n + 1) by XREAL_1:6; then
    (n + 1) in dom (Rascal(k+n)) by A0; then
    A1: (Rascal(k + n)).(n + 1) = (n + 1 - 1)*(k + n + 1 - (n + 1)) + 1
      by Def1;
    1 + 0 <= 1 + n & 0 + (n + 1) <= (k + 1) + (n + 1) by XREAL_1:6; then
    n + 1 in dom (Rascal (k + n + 1)) by A0; then
    A2: (Rascal(k + n + 1)).(n + 1) =
      (n + 1 - 1)*(k + n + 1 + 1 - (n + 1)) + 1 by Def1;
    1 + 0 <= 1 + (n + 1) & 0 + (n + 2) <= k + (n + 2) by XREAL_1:6; then
    n + 2 in dom (Rascal(k + n + 1)) by A0; then
    A3: (Rascal(k + n + 1)).(n + 2) =
      (n + 2 - 1)*(k + n + 1 + 1 - (n + 2)) + 1 by Def1;
    1 + 0 <= 1 + (n + 1) & 0 + (n + 2) <= (k + 1) + (n + 2)
      by XREAL_1:6; then
    n + 2 in dom (Rascal (k + n + 2)) by A0; then
    (Rascal(k + n + 2)).(n + 2) =
      (n + 2 - 1)*(k + n + 2 + 1 - (n + 2)) + 1 by Def1;
    hence thesis by A1,A2,A3;
  end;
