For the following proof by contradiction provide the justifications at each step, using the following equivalences and inference rules. Use the following keys:

a	Idempotent Law
b	Double Negation
c	De Morgan's Law
d	Commutative Properties
e	Associative Properties
f	Distributive Properties
g	Equivalence of Contrapositive
h	Definition of Implication
i	Definition of Equivalence
j	Identity Laws $(p \vee F = p \wedge T = p)$
k	Tautology $(p \vee \neg p = T)$
l	Contradiction $(p \wedge \neg p = F)$
m	Negation of the goal to prove
n	Modus Ponens
o	Modus Tollens
p	Transitivity of Implication
q	Conjunctive Simplification
r	Conjunctive Addition
s	Disjunctive Addition

We want to prove $s$ by a proof by contradiction from the following propositions.

$p \rightarrow q$

$r \rightarrow q$

$\neg q$

$\neg(s \wedge T) \rightarrow p$

$\neg s$ by

$\neg p$ by between $p \rightarrow q$ and $\neg q$

$s \wedge T$ by between $\neg(s \wedge T) \rightarrow p$ and $\neg p$ previously deduced.

$s$ by from $s \wedge T$

We have $s$ and $\neg s$ true, therefore we have a contradiction.