________________________________________________ Further Generalization of n-D Distance and n-D Dependent Function in Extenics Florentin Smarandache University of New Mexico, Mathematics and Science Department, 705 Gurley Ave., Gallup, NM 87301, USA e-mail：smarand@unm.edu Abstract. Prof. Cai Wen [1] defined the 1-D Distance and 1-D Dependent Function in 1983. F. Smarandache [6] generalized them to n-D Distance and n-D Dependent Function respectively in 2012 during his postdoc research at Guangdong University of Technology in Guangzhou. O. I. Şandru [7] extended the last results in 2013. Now [2015], as a further generalization, we unify all these results into a single formula for the n-D Distance and respectively for the n-D Dependent Function. Keywords: Extenics, extension distance, dependent function, attraction point, posi- tion indicator. 1 Extension Distance in 1-D Space Let’s use the notation <a, b> for any kind of closed, open, or half-closed interval { [a, b], (a, b), (a, b], [a, b) }. Prof. Cai Wen has defined the extension distance between a point x0 and a real interval S = <a, b>, by 0 ( , ) | | 2 2 o a b b a x S x       (1) where in general  : (R, R 2 ) [-(b-a)/2, +  ). (2) 2 Principle of the Extension 1-D Distance Geometrically studying this extension distance, we find the following principle that Prof. Cai has used in 1983 defining it: 0 ( , ) x S  = the geometric distance between the point x0 and the closest extremity point of the interval <a, b> to it (going in the direction that connects x0 with the optimal point O), distance taken as negative if x0  <a, b>, and as positive if x0  <a, b>.