Connected sum of simply connected manifolds is simply connected

Statement
Suppose $$n$$ is a natural number. Suppose $$M_1$$ and $$M_2$$ are fact about::simply connected manifolds (in other words, they are both manifolds that are fact about::simply connected spaces, i.e., they are both path-connected spaces with trivial fact about::fundamental group) of dimension $$n$$. Let $$M_1 \# M_2$$ denote the fact about::connected sum of these manifolds. Then, $$M_1 \# M_2$$ is also a simply connected manifold.

Similar facts

 * Fundamental group of connected sum is free product of fundamental groups in dimension at least three
 * Connected sum of compact manifolds is compact

Facts used

 * 1) uses::Point-deletion inclusion induces isomorphism on fundamental groups for manifold of dimension at least two
 * 2) uses::Seifert-van Kampen theorem