If $B$ is a lattice, then its Cartesian product with itself is also a lattice.
Now consider $[B]^2=\{(a,b):a\le b\}$, which is a subset of the Cartesian product of $B$ with itself, $B\times B$. Now the problem is how to show that $[B]^2$ is a sublattice of $B\times B$.
It is clear that $[B]^2$ is a lattice, but I have problems in showing that the $\sup$ and $\inf$ of its elements are same as in its super-set $B\times B$.
If $(a_1,b_1),(a_2,b_2)\in[B]^2$, then from $$a_1 \leq b_1 \quad\text{ and }\quad a_2\leq b_2$$ it follows that $$a_1\wedge a_2 \leq b_1\wedge b_2 \quad\text{ and }\quad a_1\vee a_2 \leq b_1\vee b_2.$$ Thus, $(a_1\wedge a_2,b_1\wedge b_2), (a_1\vee a_2,b_1\vee b_2) \in [B]^2$, these being the meet and the join of the pairs in $B^2$.