minor typo corrections again

Author: yiannis-had

content/normal-modal-logic/axioms-systems/proofs-modal-systems.tex

@@ -31,15 +31,15 @@
       \begin{derivation}
         1. & $\Log{KT5} \Proves \Diamond!A \lif \Box\Diamond!A$ & \Ax{5}\\
         2. & $\Log{KT5} \Proves !A \lif \Diamond!A$ & $\Ax{T_\Diamond}$\\
-        3. & $\Log{KT5} \Proves !A \lif \Box\Diamond!A$ & \PL.
+        3. & $\Log{KT5} \Proves !A \lif \Box\Diamond!A$ & \PL, 2, 1.
       \end{derivation}
     \item $\Log{KT5} \Proves \Ax{4}$:
       \begin{derivation}
         1. &$\Log{KT5} \Proves \Diamond\Box!A \lif \Box\Diamond\Box!A$ & \Ax{5}
         with $\Box!A$ for $p$\\
         2. & $\Log{KT5} \Proves \Box!A \lif \Diamond\Box!A$ & $\Ax{T_\Diamond}$
         with $\Box!A$ for $p$\\
-        3. & $\Log{KT5} \Proves \Box!A \lif \Box\Diamond\Box!A$ & \PL, 1, 2\\
+        3. & $\Log{KT5} \Proves \Box!A \lif \Box\Diamond\Box!A$ & \PL, 2, 1\\
         4. & $\Log{KT5} \Proves \Diamond\Box!A \lif \Box!A$ & $\Ax{5_\Diamond}$\\
         5. & $\Log{KT5} \Proves \Box\Diamond\Box!A \lif \Box\Box!A$ & \RK{}, 4 \\
         6. & $\Log{KT5} \Proves \Box!A \lif \Box\Box!A$ & \PL, 3, 5. \\
@@ -49,8 +49,8 @@
         1. & $\Log{KDB4} \Proves \Diamond\Box!A \lif !A $ & $\Ax{B_\Diamond}$ \\
         2. & $\Log{KDB4} \Proves \Box\Box!A \lif \Diamond\Box!A$ & $\Ax{D}$
         with $\Box!A$ for $p$\\
-        3. & $\Log{KDB4} \Proves \Box\Box!A \lif !A$ & \PL 1, 2\\
-        4. &  $\Log{KDB4} \Proves \Box!A \lif \Box\Box!A$ & \Ax{4} \\
+        3. & $\Log{KDB4} \Proves \Box\Box!A \lif !A$ & \PL, 1, 2\\
+        4. & $\Log{KDB4} \Proves \Box!A \lif \Box\Box!A$ & \Ax{4} \\
         5. & $\Log{KDB4} \Proves \Box!A \lif !A$ & \PL, 4, 3. \\
       \end{derivation}
     \item $\Log{KB4} \Proves \Ax{5}$: