1.  T -> [(A * R) -> (S v ~A)]
2.  ~S                          /(T * R) -> ~A
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3.  T -> [~(A * R) v (S v ~A)]                1     Material Implication
4.  T -> [(~A v ~R) v (S v ~A)]               3     De Morgan's Law
5.  T -> [(~R v ~A) v (~A v S)]               4     Commutation
6.  T -> [(~R v (~A v ~A)) v S]               5     Association
7.  T -> [(~R v ~A) v S]                      6     Tautology
8.  T -> [(~A v ~R) v S]                      7     Commutation
9.  T -> [~A v (~R v S)]                      8     Association
10. T -> [A -> (~R v S)]                      9     Material Implication
11. (T * A) -> (~R v S)                       10    Exportation
12. ~(T * A) v (~R v S)                       11    Material Implication
13. (~T v ~A) v (~R v S)                      12    De Morgan's Law
14. [(~T v ~A) v ~R] v S                      13    Association
15. (~T v ~A) v ~R                            14,2  Disjunctive Syllogism
16. ~T v (~A v ~R)                            15    Association
17. ~T v (~R v ~A)                            16    Commutation
18. (~T v ~R) v ~A                            17    Association
19. ~(T * R) v ~A                             18    De Morgan's Law
20. (T * R) -> ~A                             19    Material Implication