Mathematical Derivation of ordinary demand function:

Here we present the mathematical and more general proof of the above result. Consider, again, the initial price income situation (p₀, M₀), where x₀ is the chosen bundle. Prices change from p₀ to p₁, where p₀>p₁, and consumer's real income changes (in this case increases). Suppose we adjust consumer's money income in such a way that her  purchasing power remains the same. Consumer's money income is changed to M₁ = p₁x₀. Suppose x" is the bundle consumer buys with her adjusted money income viz., 

Since with (p₁, M₁) consumer chooses x" while x0 was available, so x" is revealed preferred to x0 and therefore was not available when x0 was chosen while (p₀, M₀) prevailed. Thus,  

Suppose only price of ith good changes, other prices remaining constant. Therefore,  

 (p_i⁰ - p_i¹) (x_i" - x_i⁰) < 0 

So, quantity change is in opposite direction of the price change. Hence, substitution effect is negative. Now, dp = (dp₁, dp₂,...,dp_i,...,d_pⁿ) and dx = (dx₁, dx₂,...,dx_i,...,dx_n)^T, so that