I'm reading Hoffman and Kunze's linear algebra book and on page 335 they state the spectral theorem for finite-dimensional inner product spaces:
Theorem 9 (Spectral Theorem). Let $T$ be a normal operator on a finite-dimensional complex inner product space $V$ or a self-adjoint operator on a finite-dimensional real inner product space $V$. Let $c_1, \dotsc, c_k$ be the distinct characteristic values of $T$. Let $W_j$ be the characteristic space associated with $c_j$ and $E_j$ the orthogonal projection of $V$ on $W_j$. Then $W_j$ is orthogonal to $W_i$ when $i \neq j$, $V$ is the direct sum of $W_1, \dotsc, W_k$, and $$ T = c_1 E_1 + \dotsb + c_k E_k. $$
Why in the beginning of the statement of this theorem can't we simply say
Let $T$ be a normal operator on a finite-dimensional complex inner product space $V$
and remove this part
or a self-adjoint operator on a finite-dimensional real inner product space $V$
since auto-adjunct operators on a finite-dimensional real inner product space is a normal one on a finite-dimensional complex inner product space.
EDIT
Answering the comments asking why an auto-adjunct operator is a normal one: If an operator $T$ is auto-adjunct, then $T^*=T$ and particularly $T^*T=TT^*$, so it's normal.