(update: I read this problem from Greenberg and Harper: Let $X=X_1\cup X_2$ and $A=X_1\cap X_2$. Using the exact sequences of triples, show that if the inclusion $(X_1,A)\hookrightarrow(X,X_2)$ induces an isomorphism in homology then the same holds for the inclusion $(X_2,A)\hookrightarrow(X,X_1)$.)
I am having a confusion over the solution given in the following MSE question: isomorphism. I cannot see how the given solution can solve the problem.
As the author suggests, I turned the problem into the one in that picture: what we have is, (at least), the arrow on the left side is an isomorphism, the two diagonals are exact. And what we try to prove is that the arrow on the right side is an isomorphism.
But as far as I can see, neither proposition (1) nor (2) in that Sum Lemma applies to the situation.
Then I have tried to prove it on my own. But to show the arrow on the right is injective, it seems to me that additional condition is needed, that is, $i_2$ is injective, which does not hold in general.
So, am I getting wrong somewhere, or is that solution not applicable to the problem? (If the latter case happens, please give me some hint to the original problem..)
By long exact sequence for triples we know that we have two short sequences $$ H_*(X_1,A) \to H_*(X,A) \to H_*(X,X_1) $$ and $$ H_*(X_2,A) \to H_*(X,A) \to H_*(X,X_2) $$ which are in fact short exact, because the connecting homomorphism vanishes since the isomorphism $ H_*(X_1,A) \to H_*(X,A) \to H_*(X,X_2)$ is a composition of a monomorphism and an epimorphism.
They fit as a "plus" into a nine-lemma diagram
$$ \require{AMScd} \begin{CD} 0 @>>> H_*(X_1,A) @>>> H_*(X_1,A) \\ @VVV @VVV @VVV \\ H_*(X_2,A) @>>> H_*(X,A) @>>> H_*(X,X_2)\\ @VVV @VVV @VVV \\ H_*(X,X_1) @>>> H_*(X,X_1) @>>> 0 . \end{CD}$$
You know that all rows and columns are short exact except from the left and right column. The nine lemma tells you the left is short exact if and only if the right is. And this is what you wanted to prove.
Concerning the proof on MSE if I am not wrong there is $H_*(X_1,A) \to H_*(X,X_1)$ contained in the diagram which either should be $a_i$ or $j_i \circ i_i$ which both are claimed to be isos but should be the zero map. Also $H_*(X,A)$ has a decomposition which I don't see. So there is some problem.