78
B. Steinbach and C. Posthoff
with the mark function of the k-fold maximum of the class C N with regard to x 0
g
re
3 (x) = max
x 0
k f
re (x),
(3.32)
and the independence function g id
3 (x) that is associated with IDM(g 3 ) satisfies
∀x 0i ∈ x 0 :
MIDC(IDM(g 3 ), x 0i ) = 0;
and all Δ-operations of f (x) with regard to x 0
g 4 (x) = Δ x 0 f (x)
belong to the class
C N 4
g
re
4 (x), g
id
4 (x)
with the mark function of the Δ-operation with regard to x 0
g
re
4 (x) = Δ x 0 f
re (x),
(3.33)
and the independence function g id
4 (x) associated with IDM(g 4 ) satisfies
∀x 0i ∈ x 0 :
MIDC(IDM(g 4 ), x 0i ) = 0.
The four independence functions are equal to each other:
g
id
1 (x) = g
id
2 (x) = g
id
3 (x) = g
id
4 (x) = g
id (x),
(3.34)
with
IDM(g) = UM(. . . UM(IDM(f ), x 01 ), . . . , x 0k )
(3.35)
and
rank(IDM(f )) + 1 ≤ rank(IDM(g)) ≤ rank(IDM(f )) + k.
If Condition (3.29) is not satisfied, we have
∀x 0i ∈ x 0 :
s min = MIDC(IDM(f ), x 0i ) = 0