Weakening of battery capacity for Watt.

@shzg yes, the greater the demand from your system the the greater the output demand on the battery. A depleted battery can then no longer supply that demand and will either shut off, or your system will "invoke" power saving measures. Replacing your battery should resolve this.

Of course you can verify this by running programs like coconutBattery and evaluating the results it shows.

Since you did not tell us which exact model MBP you are referring to, go here MacBook Pro and start selecting the proper system. All of them will have a battery changing guide and a link to replacement batteries.