Solved: Prove |HK|=|H||K| When H, K are Subgroups of G and H\capK = <e>

[chycachrrycol]

So the problem is if H and K are subgroups of G with HK = {hk $$\in$$ G| h $$\in$$ H, k $$\in$$ K}. If we know that H$$\cap$$K = <e>, show |HK|= |H||K|

My work so far:
h, j $$\in$$ H
k,l $$\in$$ K
i know that if hk = jl then j$$^{-1}$$h = lk$$^{-1}$$
But I'm not sure what to do from here.

 

[Dick]

j^(-1)h belongs to H, right? What about lk^(-1)?