you developed dyslexia at a young age? oh you mean about the car! the charging system is a relatively simple system with very few things that can actually go wrong. try checking to see if there's a large variance in voltage level between the alternator and battery while the car is running using a multimeter. the general rule is if there's a notable difference (1volt or over) you have a problem somewhere along the main charging cables (you mentioned a damaged cable....maybe its damaged at multiple points?), if the voltage is more or less the same then the likelihood is your battery being fubar'd and not excepting a charge. other worthy notes... check main earthing points around the battery (there should be a chunky one thats quite near the battery negative, make sure its solid still) you mentioned you alternator was pushing out 13.4v, thats relatively low, the norms atleast above 14v for a healthy alt, 13.4v probably drops to around the 12.4v mark when you turn your lights on which means your batteries getting feckall charge. have you added anything to the cars electrical system lately.... car alarm, remote locking, electric windows? Minimum charge voltage is 13.8, so as said with 13.4 you're putting nothing in and the battery will go flat. Ideally you want to see 14.2. You may have blown a diode in the alternator if it was shorting out.