Multivariate normality via conditional specification

If X is a k-dimensional random vector, we denote by X(i) the vector X with coordinate i deleted and by X(i,j) the vector X with coordinates i and j deleted. If for each i the conditional distribution of Xi given X(i) = x(i) is univariate normal for each x(i) [there exists]K-1 and if for each i, j the conditional distribution of Xi given X(i,j) = x(i,j) is univariate normal for each x(i,j) [there exists]k-2 then it is shown that X has a classical k-variate normal distribution.