Q. Show Additively of betas?

it is indicated earlier that any risk unique to an individual security can be removed by diversification, however as diversification increases, the amount of non market risk can be expected to decrease, but not proportionately.

For a well diversified portfolio with equal investment in each security the variance of the portfolio works out to an approximate of the variance of the market return times the square the average beta coefficient of the component securities. Since the market variance is constant for all securities the average beta becomes a measure of portfolio risk. And thus the beta as contributes to this average beta becomes a measure of risk of a security. The portfolio betas are linearly weighted combination of individual's assets beta.