In a previous memo (Computer Experiments in Finite Algebra, MAC-M-245) we described a computer system for the handling of finite groups, semigroups, subsets, finite maps, and constants. This system has been extended to read and write disk files; a mechanical procedure has been developed for extending the system; and a program (the inferential Compiler) has been written which accepts a source language consisting of mathematical statements in a standard format and compiles code which verifies these statements over a file or files of special cases (including possible counterexamples). Three limitations of the system were mentioned in the previous memo. Of these, (1) and (3) have been effectively eliminated in the current system. Limitation (2) still exists and will be overcome only in ALGEBRA III, which is briefly described in section 4.