Suppose you have a security "C" whose price at time 0 is "C(0)". In the future, in a state "i", its payoff will be "Ci". Consider a portfolio "P" consisting of "Ci" amount of each Arrow security "Ai". In the future, whatever state "i" occurs, then "Ai" pays $1 while the other Arrow securities pay $0, so "P" will pay "Ci". In other words, the portfolio "P" replicates the payoff of "C" regardless of what happens in the future. The lack of arbitrage opportunities implies that the price of "P" and "C" must be the same now, as any difference in price means we can, without any risk, (short) sell the more expensive, buy the cheaper, and pocket the difference. In the future we will need to return the short-sold asset but we can fund that exactly by selling our bought asset, leaving us with our initial profit.