Stuck on proof and need help to solve it This is not homework
I do it for fun.

I am using the SD system.
Derive $L\supset T$

$L\supset\left ( C\vee T \right ).$ Assume
$\left ( \sim L\vee B \right )\wedge\left ( \sim B\vee\sim C \right ).$ Assume

As we see from the image it is valid in SL truth tree form

There does not seem to be an easy way to derive it. I figure the the easiest method is to derive a contradiction, but how to do it after the initial set up ?

$\sim\left ( L\supset T \right ).$ Assume

For &quot;$Q$&quot; and &quot;$\sim Q$&quot; from #2, we could use $B$ and $\sim B,$ but how to separate them so they could be used ?
Any help would greatly appreaciated.
A proof by contradiction for this one will be more work than doing a conditional proof, i.e. start a subproof, assume $L$, try to get to $T$, end the subproof, and perform a Conditional Introduction to get $L \supset T$
The rest of this proof is doing the Disjunctive Syllogism pattern over and over. So, if you have the Disjunctive Syllogism rule from SD+ this should actually be quite straightforward:
\begin{array}{lll}
1. &amp; L \supset (C \lor T) &amp; Assumption\\
2. &amp; (\sim L \lor B) \&amp; (\sim B \lor \sim C) &amp; Assumption\\
3. &amp; \quad L &amp; Assumption\\
4. &amp; \quad \sim L \lor B &amp; \&amp; \ Elimination \ Premise \ 2\\
5. &amp; \quad \sim \sim L &amp; Double \ Negation \ 3\\
6. &amp; \quad B &amp; Disjunctive \ Syllogism \ 4, 5\\
13. &amp; \quad ...\\
? &amp; \quad T\\
?+1 &amp; L \supset T &amp; \supset \ Introduction \ 3-?\\
\end{array}
In SD this will be a lot more tedious., but I'll show you how to do one of these. In particular, once you have assumed $L$ as part of your conditional proof, you should be able to get to $B$ using $\sim L \lor B$:
\begin{array}{lll}
1. &amp; L \supset (C \lor T) &amp; Assumption\\
2. &amp; (\sim L \lor B) \&amp; (\sim B \lor \sim C) &amp; Assumption\\
3. &amp; \quad L &amp; Assumption\\
4. &amp; \quad \sim L \lor B &amp; \&amp; \ Elimination \ Premise \ 2\\
5. &amp; \quad \quad \sim L &amp; Assumption\\
6. &amp; \quad \quad \quad \sim B &amp; Assumption\\
7. &amp; \quad \quad \quad L &amp; Reiteration \ 3\\ 
8. &amp; \quad \quad \quad \sim L &amp; Reiteration \ 5\\ 
9. &amp; \quad \quad B &amp; \sim \ Elimination \ 6-8\\
10. &amp; \quad \quad B \ &amp; Assumption\\
11. &amp; \quad \quad B \ &amp; Reiteration \ 10\\
12. &amp; \quad B &amp; \lor \ Elimination \ 4, 5-9,10-11\\
13. &amp; \quad ...\\
? &amp; \quad T\\
?+1 &amp; L \supset T &amp; \supset \ Introduction \ 3-?\\
\end{array}
Hint: Start with your given assumptions. Assume L. Use $P\to Q ~\equiv \neg P \lor Q$ to make a chain of implications to obtain T from L.