homotopy invariance for singular homology for maps of pairs

Question

Let R be a ring, $(X,A), (Y,B)$ pairs of topological spaces and $f,g:(X,A)\to (Y,B)$ continuous maps of pairs such that there exists $H:X\times I\to Y$ homotopy with $H(A\times I)\subseteq B$, $H(x,0)=f(x)$ and $H(x,1)=g(x)$ for all $x\in X$. The claim is:
$(f,f_{|A})_*=(g,g_{|A})_*:H_n(X,A)\to H_n(Y,B)$. (we consider singular homology with coefficients in R)
I dont see the difficulty and i dont see the problem/what to prove. I know the homotopy invariance for singular homology for homotopic maps $u,v:X\to Y$. And $f,g:X\to Y$ are homotopic, therefore $H_n(f)=H_n(g)$.
But i would say that $f_{|A}, g_{|A}$ must be homotopic if you consider H restrict on A. But it can't be enough. Could you help me how to prove it? Regards