Note that "Un" cards don't always fit nicely into the general rules (that is kind of the point), so take this with a grain of salt.
With the wording on both cards, Lab Maniac still works. Lab Maniac doesn't replace losing with winning. Instead, it replaces drawing from an empty library with winning. Since Mine, Mine, Mine doesn't prevent you from drawing from an empty library (it just says you don't lose) you will try to draw a card and Lab Maniac will replace that with winning.
First, the usual warning that silver-bordered cards don't actually work within the rules.
Mine, Mine, Mine! is mostly workable as far as un-Cards go, if you translate the outdated terms ("comes into play" -> "enters the battlefield"). There's nothing there that wouldn't work if it was a real card, so let's pretend it is.
The short answer is: yes, Laboratory Maniac causes you to win the game the next time you would draw a card.
Mine! just says you don't lose from drawing a card from the empty library. It doesn't prevent the draw itself. And Laboratory Maniac's ability replaces you drawing a card with you winning, he doesn't care if you would really lose or not from that draw.
The "instead" is just used because it is a replacement ability. Since you are replacing the draw with the win, not losing with winning, it works as described.
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Laboratory Maniac
Does laboratory maniac win the game if Mine, Mine Mine is on the battlefield?
With the wording on both cards, Lab Maniac still works. Lab Maniac doesn't replace losing with winning. Instead, it replaces drawing from an empty library with winning. Since Mine, Mine, Mine doesn't prevent you from drawing from an empty library (it just says you don't lose) you will try to draw a card and Lab Maniac will replace that with winning.
Mine, Mine, Mine! is mostly workable as far as un-Cards go, if you translate the outdated terms ("comes into play" -> "enters the battlefield"). There's nothing there that wouldn't work if it was a real card, so let's pretend it is.
The short answer is: yes, Laboratory Maniac causes you to win the game the next time you would draw a card.
Mine! just says you don't lose from drawing a card from the empty library. It doesn't prevent the draw itself. And Laboratory Maniac's ability replaces you drawing a card with you winning, he doesn't care if you would really lose or not from that draw.