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In this paper, firstly we deduce the constraint of discrete Lax equations with some special forms of Lax pairs. Then to illustrate the validity of the method, we construct two discrete Lax equations and successfully obtain their constraints by the method. At the same time, infinitely many conservation laws of these two discrete equations are obtained by the discrete residue formula and the first two conserved quantities are given explicitly. It's interesting that we can obtain continuous AKNS hierarchy and KdV hierarchy through the two discrete Lax pairs. Furthermore, the 0-boundary problem for the first discrete equation is considered.