Lower bounds of sets of P-points

We show that ${\mathrm{M}\mathrm{A}}_{\kappa }$ implies that each collection of $P_{\mathfrak{c}}$-points of size at most $\kappa$ which has a $P_{\mathfrak c}$-pointasan$RK$upperboundalsohasa${P}_{\mathfrak c}$-pointasan$RK$ lower bound.