Simple--the extra energy needed (over and above electric current) comes from thermal energy. Think of it as a combined catalytic cracker/electrolysis cell without the side reactions of a full catalytic cracking process. This example was intended to point out that Boris's assertion that "...electrolysis cells are 70% efficient, tops.." was baloney. In fact, even current technology electrolysis cells AT LOW LOADS are in the mid-90% efficency region, but that drops off as the load is increased--and probably DOES drop to Boris's 70% efficiency at full production rates.
I spent years working for a chemical company that produced megatons of chlorine per year by electrolysis, so I am "somewhat" familiar with industrial-scale electrolysis. It gets REAL interesting working around a 10 kiloamp DC electric field.