I'm finding difficult to understand deduction theorem with inference rules

Question

I'm finding difficult to understand deduction theorem with inference rules. Now i'm stuck with this question because of lack of understanding in that theorem.

The problem

   1. p=>q      Premise
   2. r=>s      Premise
   3. q∨s => t  Premise
   4. ¬t        Premise
   5. p         Assumption
   6. ....       ......
   7. ....       ......
   8. ....       ......
   9. ¬p         From 4 and 8, ¬_I(Reductio ad absurdum)(Contradiction of line 
                 8 with line 4)

  10. r           Assumption
  11. ....        .....
  12. ....        .....
  13. ....        .....
  14.  ¬r          From 4 and 13, ¬_I(Reductio ad absurdum)(Contradiction of 
                   line 13 with line 4)                
  15.  ¬p∧¬r       From 9 and 14, ∧_I

Answer 1

The missing steps is that after assuming $p$ you have by $p\Rightarrow q$ that $q$ follows and then $q\lor s$ follows and by $(q\lor s)\Rightarrow t$ follows $t$. Now you have a contradiction because of $\neg t$ so you can conclude $\neg p$ instead.

In similar way you conclude $\neg r$ and $\neg p\land\neg r$ follows.

A problem seem to be is that you're mixing up terminology. The deduction theorem states that you can prove a implication by assuming the premise and proving the consequence and by that is proving that the premise implies the consequence. Or formally:

$$\begin{align}\phi&\vdash \psi\\ &\vdash \phi\Rightarrow \psi\end{align}$$

This is not that central to the problem, instead it's reductio ad absurdum which states that a statement can be proved by assuming it's opposite and proving a contradiction by which you can conclude the statement. Or formally:

$$\begin{align} \neg\psi&\vdash\phi\\ \neg\psi&\vdash\neg\phi\\ &\vdash\psi \end{align}$$