[proofplan] We verify the assertion locally in holomorphic coordinates, because the condition of having type $(a,b)$ is defined locally. In one coordinate chart, write each pure-type form as a finite sum of smooth coefficient functions times basis wedges $dz_I \wedge d\bar z_J$. Bilinearity and [associativity of the wedge product](/theorems/3559) reduce the argument to a single basis term. After reordering factors, each nonzero term has exactly $p+r$ holomorphic differentials and $q+s$ antiholomorphic differentials; terms with repeated indices vanish. [/proofplan]

[step:Expand both forms in a holomorphic coordinate chart] Let $(U,\varphi)$ be an arbitrary holomorphic coordinate chart on $M$, where $\varphi:U \to V \subseteq \mathbb C^n$ is a biholomorphism onto an [open set](/page/Open%20Set). For each $i \in \{1,\dots,n\}$, define the coordinate function $z_i:U \to \mathbb C$ by $z_i=\pi_i\circ\varphi$, where $\pi_i:\mathbb C^n\to\mathbb C$ is the $i$-th coordinate projection. Let $d\bar z_i$ denote the differential of the complex conjugate coordinate function $\bar z_i:U\to\mathbb C$. For an integer $a$ with $0\leq a\leq n$, let $\mathcal I_a$ denote the set of strictly increasing $a$-tuples $I=(i_1,\dots,i_a)$ with entries in $\{1,\dots,n\}$. For $I=(i_1,\dots,i_a)\in\mathcal I_a$, define \begin{align*} dz_I := dz_{i_1}\wedge \cdots \wedge dz_{i_a}. \end{align*} Similarly, for $J=(j_1,\dots,j_b)\in\mathcal I_b$, define \begin{align*} d\bar z_J := d\bar z_{j_1}\wedge \cdots \wedge d\bar z_{j_b}. \end{align*} Since $\alpha\in\Omega^{p,q}(M)$, by the local definition of a form of type $(p,q)$ there are smooth coefficient functions $a_{I,J}:U\to\mathbb C$, indexed by $I\in\mathcal I_p$ and $J\in\mathcal I_q$, such that \begin{align*} \alpha|_U=\sum_{I\in\mathcal I_p}\sum_{J\in\mathcal I_q} a_{I,J}\, dz_I\wedge d\bar z_J. \end{align*} Likewise, since $\beta\in\Omega^{r,s}(M)$, there are smooth coefficient functions $b_{K,L}:U\to\mathbb C$, indexed by $K\in\mathcal I_r$ and $L\in\mathcal I_s$, such that \begin{align*} \beta|_U=\sum_{K\in\mathcal I_r}\sum_{L\in\mathcal I_s} b_{K,L}\, dz_K\wedge d\bar z_L. \end{align*} [/step]

[step:Reduce the wedge product to basis terms]By bilinearity of the wedge product over smooth complex-valued functions and by associativity of the wedge product of differential forms, we have \begin{align*} (\alpha\wedge\beta)|_U=\sum_{I,J,K,L} a_{I,J}b_{K,L}\, dz_I\wedge d\bar z_J\wedge dz_K\wedge d\bar z_L. \end{align*} Here the summation ranges over $I\in\mathcal I_p$, $J\in\mathcal I_q$, $K\in\mathcal I_r$, and $L\in\mathcal I_s$.[/step]

[guided]We want to prove that $\alpha\wedge\beta$ has pure type $(p+r,q+s)$. Since this is a local assertion, it is enough to work on the arbitrary holomorphic coordinate chart $(U,\varphi)$ chosen above. The definition of $\Omega^{p,q}(M)$ says precisely that, on $U$, the form $\alpha$ is a finite smooth linear combination of basis forms with $p$ factors of the form $dz_i$ and $q$ factors of the form $d\bar z_j$. Thus \begin{align*} \alpha|_U=\sum_{I\in\mathcal I_p}\sum_{J\in\mathcal I_q} a_{I,J}\, dz_I\wedge d\bar z_J. \end{align*} The same definition applied to $\beta\in\Omega^{r,s}(M)$ gives \begin{align*} \beta|_U=\sum_{K\in\mathcal I_r}\sum_{L\in\mathcal I_s} b_{K,L}\, dz_K\wedge d\bar z_L. \end{align*} The coefficient functions $a_{I,J}:U\to\mathbb C$ and $b_{K,L}:U\to\mathbb C$ are smooth, so their products $a_{I,J}b_{K,L}:U\to\mathbb C$ are also smooth. Applying bilinearity of the wedge product to these finite sums, and then using associativity to remove unnecessary parentheses, gives \begin{align*} (\alpha\wedge\beta)|_U=\sum_{I,J,K,L} a_{I,J}b_{K,L}\, dz_I\wedge d\bar z_J\wedge dz_K\wedge d\bar z_L. \end{align*} This reduction is useful because the type of the full wedge product is now determined term by term: it remains only to inspect the elementary wedge $dz_I\wedge d\bar z_J\wedge dz_K\wedge d\bar z_L$.[/guided]

[step:Reorder each nonzero basis term into pure type] Fix indices $I\in\mathcal I_p$, $J\in\mathcal I_q$, $K\in\mathcal I_r$, and $L\in\mathcal I_s$. If an index occurs both in $I$ and in $K$, then $dz_i$ appears twice in $dz_I\wedge dz_K$, so the corresponding wedge term is zero by alternatingness. If an index occurs both in $J$ and in $L$, then $d\bar z_j$ appears twice in $d\bar z_J\wedge d\bar z_L$, so the corresponding wedge term is again zero. Assume now that $I$ and $K$ are disjoint and that $J$ and $L$ are disjoint. Let $I\cup K$ denote the unique strictly increasing $(p+r)$-tuple obtained by ordering the union of the entries of $I$ and $K$. Let $J\cup L$ denote the unique strictly increasing $(q+s)$-tuple obtained by ordering the union of the entries of $J$ and $L$. Reordering the factors moves the $r$ holomorphic factors $dz_K$ past the $q$ antiholomorphic factors $d\bar z_J$, and then orders the holomorphic and antiholomorphic blocks. Therefore there is a sign $\varepsilon_{I,J,K,L}\in\{-1,1\}$ such that \begin{align*} dz_I\wedge d\bar z_J\wedge dz_K\wedge d\bar z_L=\varepsilon_{I,J,K,L}\, dz_{I\cup K}\wedge d\bar z_{J\cup L}. \end{align*} The right-hand side is a basis form with exactly $p+r$ holomorphic differentials and exactly $q+s$ antiholomorphic differentials. [/step]

[step:Conclude the local expression has type $(p+r,q+s)$] Substituting the preceding termwise description into the local expansion of $(\alpha\wedge\beta)|_U$, every nonzero summand is a smooth coefficient function times a basis element $dz_A\wedge d\bar z_B$ with $A\in\mathcal I_{p+r}$ and $B\in\mathcal I_{q+s}$. Hence, on the arbitrary chart $U$, \begin{align*} (\alpha\wedge\beta)|_U\in\Omega^{p+r,q+s}(U). \end{align*} Since membership in $\Omega^{p+r,q+s}$ is local in holomorphic coordinate charts, this proves \begin{align*} \alpha\wedge\beta\in\Omega^{p+r,q+s}(M). \end{align*} [/step]