reserve a,b,n for Element of NAT;

theorem
  for n being Nat holds 3 * Fib(n) + Lucas(n) = 2 * Fib(n+2)
proof
  defpred P[Nat] means 3*Fib($1)+Lucas($1)=2*Fib($1+2);
  let n be Nat;
  0+1+1=2;
  then
A1: Fib(2) = 1 by PRE_FF:1;
A2: for k being Nat st P[k] & P[k+1] holds P[k+2]
  proof
    let k be Nat;
    assume
A3: ( P[k])& P[k+1];
    3*Fib(k+2)+Lucas(k+2)= 3*(Fib(k)+Fib(k+1))+Lucas(k+2) by FIB_NUM2:24
      .= 3*Fib(k)+3*Fib(k+1)+(Lucas(k)+Lucas(k+1)) by Th12
      .= 2*Fib(k+2)+2*Fib((k+1)+2) by A3
      .= 2*(Fib(k+2)+Fib((k+2)+1))
      .= 2*Fib((k+2)+2) by FIB_NUM2:24;
    hence thesis;
  end;
  0+1+1+1=3;
  then Fib(3)=2 by A1,PRE_FF:1;
  then
A4: P[1] by Th11,PRE_FF:1;
A5: P[0] by A1,Th11,PRE_FF:1;
  for k being Nat holds P[k] from FIB_NUM:sch 1 (A5, A4, A2);
  hence thesis;
end;
