In the case of m+n variables we have n equations $\displaystyle F_i(x_1,...,x_m;y_1,...y_n)=0$ which must be satisfied for $\displaystyle x_i=x^0_i, y_j=y^0_j$ for the theorem to be applicable, althought (I understand that) there might be more than one solutions. But if for a given $\displaystyle x_i=x^0_i$ there are more than one $\displaystyle y_j=y^k_j$ satisfying the n equations (and I do not find any statement in the theorem preventing this) how can the theorem ensure that there is one and only one set of solutions $\displaystyle y_j=f_j(x_1,...,x_m)$ which are continuous, satisfy $\displaystyle F_j=0$ and for which $\displaystyle y^0_j=f_j(x^0_1,...,x^0_m)$ ? I mean how can f be a function?...Of course I must be missing something...

thanks a lot