Answer
The property was verified for $n=1$.
The property was verified when $n$ was changed by $n+1$.
Work Step by Step
Prove the property for $n=1$.
$x(y_1)=xy_1$
The property is verified for $n=1$.
Assuming that $x(y_1+y_2+...+y_n)=xy_1+xy_2+...+xy_n$ for all integers $n\geq1$, we need to prove that $x(y_1+y_2+...+y_n+y_{n+1})=xy_1+xy_2+...+xy_n+xy_{n+1}$:
$x(y_1+y_2+...+y_n+y_{n+1})=x(y_1+y_2+...+y_n)+xy_{n+1}=xy_1+xy_2+...+xy_n+xy_{n+1}$
