Cohomologies and analytic differential forms

A proof that homology groups $H^k(X;\mathscr O_X)$ of aЁcomplex analytic space $X$, countable at infinity and locally smoothly contractible, with coefficients in the lattice bundle $\mathscr O_X$, are canonically isomorphic to the corresponding homology groups $H^k\Gamma(X;\matscr A_X^{0,*})$ of the finite complex of analytic differential forms $\Gamma(X\mathscr A_X^{0,*})$ with the exterior differential $d''$ as a coboundary operator.