Steenrod originally defined his squares using explicit cochain-level formulas for simplicial mod-2 cochains. To this end, he introduced *higher cup products*, which control the failure of the usual cup product to be supercommutative on the cochain level. On the other hand, H. Cartan formula tells us how Steenrod squares relate to the usual cup product of mod-2 cohomology classes. 

**Questions.**

(1) How does one show this using Steenrod's definition of Steenrod squares? 

(2) Is there some cochain-level relation between cup products of various degrees which implies 

Cartan's formula?