The Guiming people was originally an ethnic minority in a southwestern vassal state, under the luring and winning-over policy of the Song Dynasty, they came over and pledged allegiance.

On this basis, a method of proof by induction is shown in order-sorted specification.

In this paper the authors introduced the implemental techniques of transformation equational specification into order-sorted equational specification, the proof techniques of inductive theorems on the order-sorted equational specification based on specification transformation for proof by induction.