Proof That 1 - Forms are Semi - Distributive

The 1 - form is semi distributive that is, if  
\[\omega_1 , \: \omega_2 , \: \omega_3\]
  are 1 - forms ans  
\[f,\: g\]
  are functions, then  
\[(f \omega_1 + g \omega_2) \wedge \omega_3 = f \omega_1 \wedge \omega_3 + g \omega_2 \wedge \omega_3 \]
\[f, \: g\]
  be real valued functions on a domain  
\[D \subseteq \mathbb{R}^n\]
  and let  
\[\omega_1 ,\: \omega_2 ,\: \omega_3\]
  be 1 - forms defined on  
\[D \]
&. Then
\[\omega_{1 \mathbf{x}} =f_1 (\mathbf{x}) dx_1 +...+ f_n (\mathbf{x}) dx_n \]

\[\omega_{2 \mathbf{x}} =g_1 (\mathbf{x}) dx_1 +...+ g_n (\mathbf{x}) dx_n \]

\[\omega_{3 \mathbf{x}} =h_1 (\mathbf{x}) dx_1 +...+ h_n (\mathbf{x}) dx_n \]

Expanding the left hand side gives
\[\begin{equation} \begin{aligned} & ( \omega_{1 \mathbf{x}} + \omega_{2 \mathbf{x}}) \wedge \omega_3 \\ &=(f(f_1 (\mathbf{x}) dx_1 +...+ f_n (\mathbf{x}) dx_n)+ g(g_1 (\mathbf{x}) dx_1 +...+ g_n (\mathbf{x}) dx_n)) \wedge (h_1 (\mathbf{x}) dx_1 +...+ h_n (\mathbf{x}) dx_n ) \\ &= \sum_{i=1}^n \sum_{j=1}^n ( h_i (ff_i +gg_i) -h_i (ff_j +gg_j)) dx_i \wedge dx_j \end{aligned} \end{equation}\]

Expanding the right hand side gives
\[\begin{equation} \begin{aligned} & f \omega_1 \wedge \omega_3 + g \omega_2 \wedge \omega_3 \\ &= f(f_1 (\mathbf{x}) dx_1 +...+ f_n (\mathbf{x}) dx_n) \wedge (h_1 (\mathbf{x}) dx_1 +...+ h_n (\mathbf{x}) dx_n)\\ &+ g(g_1 (\mathbf{x}) dx_1 +...+ g_n (\mathbf{x}) dx_n) \wedge (h_1 (\mathbf{x}) dx_1 +...+ h_n (\mathbf{x}) dx_n) \\ &=\sum_{i=1}^n \sum_{j=1}^n (h_jff_i -h_i ff_j) dx_i \wedge dx_j +\sum_{i=1}^n \sum_{j=1}^n (h_jgg_i -h_igg_j)dx_i \wedge dx_j \\ &= \sum_{i=1}^n \sum_{j=1}^n ( h_i (ff_i +gg_i) -h_i (ff_j +gg_j)) dx_i \wedge dx_j \end{aligned} \end{equation}\]

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