Simon Cowell -- First Pictures of Baby ... Hidden Inside Friend's Wife

Here it is ... THE FIRST PHOTO OF SIMON COWELL'S ALLEGED BABY -- unfortunately, you can't really see the thing because it's buried inside the womb of his friend's wife.

Simon's alleged baby mama Lauren Silverman -- the 36-year-old wife of his real estate pal Andrew Silverman -- stepped out in the Hamptons on Thursday, carrying Simon's little bundle of rapidly multiplying cells in her uterus (allegedly).

As we reported, Andrew just filed divorce docs claiming Lauren committed adultery when she conceived Simon's baby.

Adultery ... the real "ex" factor.