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You always need to have in mind that $A$ is a hypothetical algorithm, since our goal in the reduction is to contradict the existence of such an efficient $A$. Now to your concrete security framework: Here, you are not satisfied by the fact that a hypothetical poly-time $A$ implies a poly-time reduction $A'$, but your aim is that the reduction does not take ...
The equation $t'=t+n\cdot t_c$ is an estimation to put an upper limit on $t'$. It might be possible that an attacker $A'$ can use a different, more efficient algorithm. But since the attack will work with using $A$, there exists an attacker $A'$ with at most $t'$. This means it's actually not an equation, but an inequality $t' \leq t + n \cdot t_c$. And then ...