A two-dimensional grid grammar was designed to fill up the missing ring in the Equivalence Theorem for recognizable picture languages (REC), summarized in a survey paper by Giammarresi and Restivo. Instead of 2-dimensional on-line tessellation automata, grid automata were introduced, which were closer to the traditional binary tree automata, to bridge the grid grammar and other approaches of describing the class of REC. Meanwhile the standard (existential) monadic second order logic was substituted by a weaker logic framework: positive monadic partition logic. A new and complete version of Equivalence Theorem for REC is presented.