reserve a,b,n for Element of NAT;

theorem
  for b,n being Element of NAT holds GenFib(0,b,n) = b * Fib(n)
proof
  let b,n be Element of NAT;
  defpred P[Nat] means GenFib(0,b,$1)=b*Fib($1);
A1: P[1] by Th32,PRE_FF:1;
A2: for k being Nat st P[k] & P[k+1] holds P[k+2]
  proof
    let k be Nat;
    assume that
A3: P[k] and
A4: P[k+1];
    GenFib(0,b,k+2)=b*Fib(k)+GenFib(0,b,k+1) by A3,Th34
      .=b*(Fib(k)+Fib(k+1)) by A4
      .=b*Fib(k+2) by FIB_NUM2:24;
    hence thesis;
  end;
A5: P[0] by Th32,PRE_FF:1;
  for k being Nat holds P[k] from FIB_NUM:sch 1 (A5, A1, A2);
  hence thesis;
end;
