State and prove addition theorem on probability

Statement : If A&B are any two events in a sample space of random experiment then

$P(A\cap B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$

Case (i) : Suppose A & B are mutually exclusive events

$\Rightarrow A\cap B=\varphi \Rightarrow P(A\cap B)=P\left(\varphi \right)=0$

From axiam of additivity  $P(A\cup B)=P\left(A\right)+P\left(B\right)=P\left(A\right)+P\left(B\right)-0$

$=P\left(A\right)+P\left(B\right)-P\left(\varphi \right)=P\left(A\right)+P\left(B\right)-P(A\cap B)$

Case (ii) : Suppose A&B are not mutually exclusive 

$\Rightarrow A\cap B\ne \varphi \Rightarrow A\cup B=A\cup (B-A)$

$\therefore P(A\cup B)=P(A\cup (B-A\left)\right)=P\left(A\right)+P(B-A)$

By axiam of additivity  A and B-A are disjoints

$=P\left(A\right)+P[B-(A\cap B\left)\right]$

$=P\left(A\right)+P\left(B\right)-P(A\cap B)[\because A\cap B\subseteq B]$

$\therefore P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$