No doubt re-inventing wheel, but the explanations of double induction online seem quite confusing to me. I'm not sure how widely applicable the following may be, but you may find the following analogy to ordinary induction to be useful.
With ordinary induction, we want to prove that for all x in N, P(x) where P is a unary predicate.
With double induction, we want to prove that for all x, y in N, P(x,y) where P is a binary predicate.
1. Base case:
Ordinary induction: Prove P(1)
Double induction: Prove P(1,1)
2. Inductive step:
Ordinary induction: For x in N, suppose P(x) and prove P(x+1)
Double induction: For x, y in N, suppose P(x,y) and prove P(x+1,y) and P(x,y+1).