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problem solved itself, sorry for your inconvenience. I'll try to post better questions next time.

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There is a paper, which is more general than Gaussian distributions.

Bohrnstedt, George W., and Arthur S. Goldberger. "On the exact covariance of products of random variables." Journal of the American Statistical Association 64.328 (1969): 1439-1442.

https://www.jstor.org/stable/pdf/2286081.pdf?casa_token=m1_rSG4hiwIAAAAA:sGjisD6dekanCat5EBGl0yvdn1EJ5LY783fq4VKdxNhd0RA4z4izkcyDJ6yU7MylDke_qD335SWUNu1gJ5opTzXFWIVQWDrjoMvxcFgvMBEgG6r8En0W

Their equation (9) $$ V(xy) = E^2(x)V(y) + E^2(y)V(x) + V(x)V(y). \quad (9) $$ reduces to your result

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