Translation:Sometimes after the rain there is a rainbow.
"Sometimes there is the rainbow after rain." - Why should it be "the" rainbow? Is ther one particular rainbow that comes after rain, while all other sorts don't? Sounds weird.
It doesn't have to be 'the', the main translation is "Sometimes after the rain there is a rainbow."
"Sometimes there is a rainbow after rain." is accepted as well.
Sorry, the Polish sentence is factually incorrect.
I'd accept "Sometimes with the rain there is a rainbow" - but that doesn't translate po. A rainbow needs Sun and rain (or other water droplets, e.g. a waterfall) at the same time: it's simple school physics.
Of course, it might have stopped raining where I am: if it's still raining somewhere between me and the Sun, I'll still see the rainbow.