reserve n, k, r, m, i, j for Nat;

theorem Th24:
  for n being Nat holds Fib (n + 2) = Fib (n) + Fib (n + 1)
proof
  defpred P[Nat] means Fib($1 + 2) = Fib($1) + Fib($1 + 1);
  let n be Nat;
A1: for k being Nat st P[k] & P[k+1] holds P[k+2]
  proof
    let k be Nat;
    assume P[k];
    assume P[k+1];
    Fib (k+2+2) = Fib ((k+2+1)+1) .= Fib (k+2) + Fib (k+2+1) by PRE_FF:1;
    hence thesis;
  end;
  Fib (0+2) = Fib (0+1+1) .= Fib (0) + Fib (1) by PRE_FF:1;
  then
A2: P[0];
A3: P[1] by PRE_FF:1;
  for n being Nat holds P[n] from FIB_NUM:sch 1(A2,A3,A1);
  hence thesis;
end;
