I understand Lagrange multiplier for one equation constraint task. But what about several equations formal proof of Lagrange multipliers method?
My own non-formal proof for $R3$ case:
Suppose we have $f(x,y,z)$ and 2 constraints $g(x,y,z) = 0$ and $h(x,y,z) = 0$. These 2 constraints are some surfaces in $R3$ space intersecting somewhere and so forming some curve. Gradients of $g$ and $h$ in any point on the curve forming some plane. It is obvious that function $f$ will not change only in case when gradient of objective function $f$ will be in the plane which is formed by gradients of g and h (vector lies in the plane only if it is a linear combination of at least 2 vectors lie in the plane).
So the initial requirement of gradients collinearity transforms in this case to requirement to be in the same hyperplane with all constraint gradients.