As the expansion of the falsity- and truth-preserving principles, in Section 7.1, the general principle of quotient space approximation method is presented. In Section 7.2, its relation to the second-generation wavelet multi-resolution analysis is discussed. So quotient space approximation can seek out new mathematical tools from wavelet analysis. It is also the expansion of information synthesis in some sense. In Section 7.3, the relation between quotient space approximation method and fractal geometry is discussed. In Section 7.4 the theory is extended to structures induced from closure operations and the tolerance relations. We also show that many useful properties in traditional theory are still available under the expansion. As an application of tolerant relation-based quotient space theory, a protein structure prediction task is presented.