The Attempt at a Solution
So I already proved that I+J=R, so that there are x in I and y in J such that x+y=1. Then, f(x)=(I, 1+J) because x is in I and x=1-y which is in 1+B. Similarly, f(y)=(1+I, J). Now, consider kx+ry. f(kx+ry)=f(k)f(x)+f(r)f(y)=(k+I, k+J)(I, 1+J)+(r+I, r+J)(1+I, J)=(I, k+J)+(r+I)(J)=(r+I)(k+J). Is that right?