You know that different boundary value problems with very simple local conditions for Cauchy-Riemann system were investigated before. Here we’ll try to find sufficient conditions for the fredholmness of boundary value problem with general non-local boundary conditions for Cauchy-Riemann equation (it is an elliptic equation of the first order). The domain on which we’ll search the problem is multi connected. Boundary conditions are given as the linear combinations of the values of unknown function that it accepts at the points of intersection of the domain border and the perpendicular drawn through an arbitrary point of x-axis.