reserve a,b,n for Element of NAT;

theorem
  for n being Element of NAT holds 2 * Fib(2*n+1) = Lucas(n+1) * Fib(n)
  + Lucas(n) * Fib(n+1)
proof
  defpred P[Nat] means 2*Fib(2*$1+1)= Lucas($1+1)*Fib($1)+Lucas($1)*Fib($1+1);
  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 that
A3: P[k] and
A4: P[k+1];
    set f2 = Fib(k+2);
    set f1 = Fib(k+1);
    set f = Fib(k);
    set l2 = Lucas(k+2);
    set l1 = Lucas(k+1);
    set l = Lucas(k);
A5: 2*Fib(2*k)=2*(Fib(2*k+2)-Fib(2*k+1)) by FIB_NUM2:30
      .=2*(Fib((2*k+1)+1)-Fib(2*k+1))
      .=2*((Fib((2*k+1)+2)-Fib(2*k+1))-Fib(2*k+1)) by FIB_NUM2:29
      .=2*Fib(2*k+3)-2*(2*Fib(2*k+1))
      .=l2*f1 + l1*f2 - 2*(Lucas(k+1)*Fib(k)+Lucas(k)*Fib(k+1)) by A3,A4
      .=l2*f1 + l1*f2 - 2*l1*f - 2*l*f1
      .=(l+l1)*f1 + l1*f2 - 2*l1*f - 2*l*f1 by Th12
      .=l*f1 + l1*f1 + l1*(f+f1) - 2*l1*f - 2*l*f1 by FIB_NUM2:24
      .=2*l1*f1 - l*f1 - l1*f;
    2*Fib(2*(k+2)+1)=2*(Fib((2*k+3)+2))
      .=2*(Fib(2*k+3)+Fib((2*k+3)+1)) by FIB_NUM2:24
      .=2*(Fib(2*k+3)+Fib((2*k+2)+2))
      .=2*(Fib(2*k+3)+(Fib(2*k+2)+Fib((2*k+2)+1) ) ) by FIB_NUM2:24
      .=2*(Fib(2*k+3)+Fib((2*k+1)+1)+Fib(2*k+3))
      .=2*(Fib(2*k+3)+(Fib((2*k+1)+2)-Fib(2*k+1))+Fib(2*k+3)) by FIB_NUM2:29
      .=3*(2*Fib(2*k+3))-2*Fib(2*k+1)
      .=3*(Lucas(k+2)*Fib(k+1)+Lucas(k+1)*Fib(k+2))-2*Fib(2*k+1) by A4
      .=3*l2*f1+3*l1*f2-l1*f-l*f1 by A3
      .=3*(l+l1)*f1+3*l1*f2-l1*f-l*f1 by Th12
      .=(3*l+3*l1)*f1+3*l1*(f+f1)-l1*f-l*f1 by FIB_NUM2:24
      .=3*l*f1 + 4*l1*f1 + 3*l1*f + (2*l1*f1 - l*f1 - l1*f)
      .=3*l*f1 + 4*l1*f1 + 3*l1*f + 2*(l*f) by A5,Th28
      .=l*f1+l1*f1+l1*(f+f1)+2*l*f+2*l*f1+2*l1*f+2*l1*f1
      .=(l+l1)*f1+l1*f2+2*l*f+2*l*f1+2*l1*f+2*l1*f1 by FIB_NUM2:24
      .=l2*f1+l1*f2+2*l*f+2*l*f1+2*l1*f+2*l1*f1 by Th12
      .=Lucas(k+2)*Fib(k+1)+Lucas(k+1)*Fib(k+2)+2*(l*f+l*f1+l1*f+l1*f1)
      .=2*Fib(2*k+3)+2*(l*f+l*f1+l1*f+l1*f1) by A4
      .=2*(Fib(2*k+3)+(Lucas(k)+Lucas(k+1))*(Fib(k)+Fib(k+1)))
      .=2*(Fib(2*k+3)+(Lucas(k)+Lucas(k+1))*Fib(k+2)) by FIB_NUM2:24
      .=2*(Fib(2*k+3)+Lucas(k+2)*Fib(k+2)) by Th12
      .=2*Fib(2*k+3)+Lucas(k+2)*Fib(k+2)+Lucas(k+2)*Fib(k+2)
      .=(Lucas(k+1)*Fib(k+2)+Lucas(k+2)*Fib(k+1))+ Lucas(k+2)*Fib(k+2)+Lucas
    (k+2)*Fib(k+2) by A4
      .=(Lucas(k+1)+Lucas(k+2))*Fib(k+2)+Lucas(k+2)*(Fib(k+1)+Fib(k+2))
      .=(Lucas(k+1)+Lucas((k+1)+1))*Fib(k+2)+Lucas(k+2)*Fib((k+1)+2) by
FIB_NUM2:24
      .=Lucas((k+2)+1)*Fib(k+2)+Lucas(k+2)*Fib((k+2)+1) by Th12;
    hence thesis;
  end;
  1+1+1=3;
  then 2*Fib(2*1+1)=2*2 by A1,PRE_FF:1
    .=Lucas(1+1)*Fib(1)+Lucas(1)*Fib(1+1) by A1,Th11,Th14,PRE_FF:1;
  then
A6: P[1];
A7: P[0] by Th11,PRE_FF:1;
  for k being Nat holds P[k] from FIB_NUM:sch 1 (A7, A6, A2);
  hence thesis;
end;
