reserve a,b,n for Element of NAT;

theorem
  for n being Element of NAT holds 2 * Lucas(n+2) = Lucas(n) + Lucas(n+3 )
proof
  let n be Element of NAT;
A1: Lucas(n+1) + Lucas(n+2) = Lucas(n+3) by Th13;
  thus 2 * Lucas(n+2) = Lucas(n+2) + Lucas(n+2)
    .= Lucas(n+1) + Lucas(n) + Lucas(n+2) by Th12
    .= Lucas(n) + Lucas(n+3) by A1;
end;
