This is from Grimmett and Stirzaker, Chapter 1, page 6.

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> The triple $(\Omega,\mathscr{F},P)$ is a probability space. It has some important properties.
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> (a) $P(A^C)=1-P(A)$
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> (b) if $B\supseteq{A}$ then $P(B)=P(A)+P(B\setminus{A})\ge{P(A)}$
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> (c) $P(A\cup{B})=P(A)+P(B)-P({A}\cap{B})$
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The proofs for (b) and (c) are not entirely clear to me.

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> (b) $B={A}\cup({B}\setminus{A})$. This is the union of disjoint sets and therefore,
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> $P(B)=P(A)+P({B}\setminus{A})$
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I get the above proof. Moving onto the proof for Lemma(c)...

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> (c) $A\cup{B}=A\cup({B}\setminus{A})$, which is a disjoint union. Therefore by (b),
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> $P({A}\cup{B})=P(A)+P(B\setminus{A})=P(A)+P({B}\setminus({A}\cap{B}))=P(A)+P(B)-P({A}\cap{B})$
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I don't get how (c) is being deduced. Moreover, the start of the proof didn't make sense. Any enlightenment would be appreciated!