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The variance of the portfolio is $$V_p=\sum_i \sum_j w_i w_j Cov(r_i,r_j)$$ because of the properties of $Cov(\cdot,\cdot)$ we can rewrite this as $$V_p=\sum_i w_i Cov(r_i,\underbrace{\sum_j w_j r_j}_{R_P})$$ where $R_p$ is the return on the portfolio. So we have $$V_p=\sum_i w_i Cov(r_i,R_p)$$ QED (And it is true of every portfolio, not just the ...