Just a hint of how I could prove this...

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Prove that if z1z2 = 0 then z1 = 0 or z2 = 0 or both.

thanks in advance
 
We are given z1z2=0\displaystyle z_1 \cdot z_2 = 0. If z2=0\displaystyle z_2 = 0 then there is nothing more to prove.

So suppose that z20\displaystyle z_2 \not= 0, then from the field properties of real numbers 1z2\displaystyle \frac{1}{{z_2 }} the multiplicative inverse exists.
And
\(\displaystyle \begin{array}{l}
0 = z_1 z_2 \\
0\left( {\frac{1}{{z_2 }}} \right) = z_1 z_2 \left( {\frac{1}{{z_2 }}} \right) \\
0 = z_1 \\
\end{array}\),
we are done.
 
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