Using the notion of inductive proper
q
−
1
−
LCC
q - 1 - {\text {LCC}}
introduced in this note, we will prove the following theorems. Theorem 1. Let
M
M
be an
R
∞
{R^\infty }
-manifold and let
H
:
X
×
I
→
M
H:X \times I \to M
be a homotopy such that
H
0
{H_0}
and
H
1
{H_1}
are
R
∞
{R^\infty }
-deficient embeddings. Then, there is a homeomorphism
F
F
of
M
M
such that
F
∘
H
0
=
H
1
F \circ {H_0} = {H_1}
. Moreover, if
H
H
is limited by an open cover
α
\alpha
of
M
M
and is stationary on a closed subset
X
0
{X_0}
of
X
X
and
W
0
{W_0}
is an open neighborhood of
\[
H
[
(
X
−
X
0
)
×
I
]
i
n
M
,
H[(X - {X_0}) \times I] \quad {in\;M,}
\]
then we can choose
F
F
to also be
St
4
(
α
)
\operatorname {St}^4(\alpha )
-close to the identity and to be the identity on
X
˙
0
∪
(
M
−
W
0
)
\dot X_{0} \cup (M - {W_0})
. Theorem 2. Every closed, locally
R
∞
(
Q
∞
)
{R^\infty }({Q^\infty })
-deficient subset of an
R
∞
(
Q
∞
)
{R^\infty }({Q^\infty })
-manifold
M
M
is
R
∞
(
Q
∞
)
{R^\infty }({Q^\infty })
-deficient in
M
M
. Consequently, every closed, locally compact subset of
M
M
is
R
∞
(
Q
∞
)
{R^\infty }({Q^\infty })
-deficient in
M
M
.