So, condition number one. When we substitute into all these functions the coordinates, x coordinates, all the point, we get y value and that's true for all j from one to m inclusive. Secondly, if we turn to the system, our original system of equations, when I'm taking jth equation and substitute for y values, the implicit functions, I get zero, identically zero. So, this is true for all x which belongs to the [inaudible] , B tilde. Thirdly, all such derivatives can be found. This phrase means that the formulas are derived but they're kind of cumbersome and I'll show you using some examples that, in each particular case, such a derivative can be easily found, more or less easily found. Let's proceed with the examples from, first, in microeconomics. Let us consider an IS-LM model where we have two equilibrium equations or conditions. There is an equilibrium in the goods market and an equilibrium in money market. So, we have a system of two equations, the outputs in an economy, y, equals the consumption, and this consumption function is the function of the output less of taxes imposed, so this is a disposable income, plus the investment, investment is a function of the interest rate, and plus the government expenditures. So, this equation defines equilibrium in the goods market. The second equation concerns equilibrium in the money markets. Money supply should equal demand on money, which is a function of the output, y, and the interest rate, r. The properties of the functions involved are known. For instance, it is known that the derivative of the consumption function lies between zero and one and it is called marginal propensity to consume. As for the investment function, its derivative is negative. Demand and money, derivative of this function with respect to the output, is positive and derivative with respect to the interest rate is negative. Now, let's consider a comparative statics problem. Let government expenditures be fixed and the money supply will be also fixed and T taxes will change. That means that dT increment in T, in taxes, is not zero but the increments in money supply and government expenditures are zero. How will the change of taxes will affect the output and the interest rate? So, we're interested in finding expressions for two derivatives: the derivative of the output with respect to taxes and derivative of the interest rate with respect to taxes, so that's the question. First of all, we need to check whether this third IFT is applicable. In order to do that, let us consider Jacobian of two equations. Remember, the Jacobian is a matrix, two by two matrix whose rows are made of derivatives. So, we need to differentiate the first equation with respect to Y and r which are endogenous variables and we fill in the first row of this Jacobian matrix. Now, we proceed with the second equation and also we differentiate with respect to Y and interest rate r and fill in the entries in the second row. After that, we need to check that this square, two by two matrix, is a non-singular matrix, so let us do it. So, J, this is Jacobian, is a matrix. First of all, let us move all these three terms to the left and differentiate with respect to Y, then I get one minus C prime. After that, in order to fill in this missing entry in the first row, I would like to remind you that this term was already moved to the left, now we'll differentiate with respect to r. That's how we get minus I prime. Now, here, we simply differentiate with respect to the output and with respect to the interest rate. Somewhere here, I will calculate the determinant of this matrix, it equals. Okay, let's check the sign. This difference within the brackets is positive, this derivative is negative and this derivative is also negative and this derivative is positive. So, all in all, we have negative result which tells us that IFT is applicable and the only thing is we need to find out the expressions for these derivatives.