Suppose a sum f+′g of maps f,g:(In,∂In)→(X,x0) is defined using a coordinate of In other than the first coordinate as in the usual sum f+g. Verify the formula (f+g)+′(h+k)=(f+′h)+(g+′k), and deduce that f+′k≃f+k so the two sums agree on πn(X,x0), and also that g+′h≃h+g so the addition is abelian.