The regulation for proposition proving based on the accumulation principle was discussed, and the proofs of the ensted interval theorem, the supremum theorem and the finite covering theorem were put forward here in term of above regulation.

Interval equivalence classes and interval function S-rough sets are defined based on interval attribute of function S-rough sets,and the finite covering theorem of closed interval equivalence classes and that of closed interval function S-rough sets are put forward.

In this paper,both the finite-covers technique and the radial point-interpolation method are integrated together to develop an element-free radial point-interpolation procedure that is based on finite covers technique which takes both advantages of these two types of numerical methods.

The finite cover is the essential technique in this method.

The technique was based on finite covering and partition of unity.

In the paper, by the use of concepts of finite cover and partition of unit in manifold ideas, cover function and partition of unit function are established on finite covers so that approximation of displacement field function is got.