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Let $T$ be an NSOP$_1$ theory. Recently I. Kaplan and N. Ramsey proved that in $T$, the so-called Kim-independence ($\phi(x,a_0)$ Kim-divides over $A$ if there is a Morley sequence $a_i$ such that $\{\phi(x,a_i)\}_i$ is inconsistent) satisfies nice properties over models such as extension, symmetry, and type-amalgamation. In a joint work with J. Dobrowolski and N. Ramey we continue to show that in $T$ with nonforking existence, Kim-independence also satisfies the properties over any sets, in particular, Kimâ s lemma, and 3-amalgamation for Lascar types hold. Modeling theorem for trees in a joint paper with H. Kim and L. Scow plays a key role in showing Kimâ s lemma. If time permits I will talk about a result extending the non-finiteness (except 1) of the number of countable models of supersimple theories to the NSOP$_1$ theory context.