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Since for the neighbourhood W k we have ω(x) = z k then Fr ∩ W k ⊂ W0k−1 . (8) Let H ⊂ W0k−1 be the set of all nonproper points of the mapping ϕ : W0k−1 → E l . It is easy to see that H ∩ W0k−1 ⊂ H [see (6)] and, since Fr ∩ W1k ⊂ H then it follows from (8) that Fr ∩ W1k ⊂ H . By virtue of the induction hypothesis, the set ϕ(H ) has ﬁrst category in E l . Since Fr ∩ W1k ⊂ H the set ϕ(Fr ∩ W1k ) is nowhere dense in E l . Thus we have completed the proof for the case r > 0. Thus, Theorem 4 is proved when M k has no boundary.

Uk , ξ1 , . . , ξ k . Since the ﬁrst component of the vector u in the space C 2k equals one [see (30)], the coordinates of the row u∗ in the manifold S 2k−1 can be set to be the remaining components v 2 , . . , v 2k of the 3rd April 2007 9:38 WSPC/Book Trim Size for 9in x 6in 38 main L. S. Pontrjagin vector u in the space C 2k . In the chosen coordinates, the mapping τ is written (according to (30)) as v i = ui , i = 2, . . , k; v k+j = ∂ϕj (x) ∂ξ 1 k ui + i=1 ∂ϕj (x) ∂ξ i (31) , j = 1, . .

K; j = 1, . . , q + 1 − k. (13) Let (x, u∗ ) be a point of the manifold N q close to the point (a, u∗ ) = (a, e∗q+1 ). On the ray u∗ , let us choose a vector u satisfying the condition (u, eq+1 ) = 1. Denote the remaining q components of the vector u in the basis e1 , . . , eq+1 by u1 , . . , uq : ui = (u, ei ), i = 1, . . , q. The orthogonality condition for the vector u and h(M k ) at the point h(x) now looks like 0= ∂h(x) u, ∂ξ i q−k i uk+j =u + j=1 ∂ϕj (x) ∂ϕq+1 (x) + , i = 1, . .