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<b>let </b><font color="Maroon">GF</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">V</font> be  <a href="matrlin.html#V2">finite-dimensional</a> <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">GF</font>;<br/><b>let </b><font color="Maroon">W1</font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">W2</font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/>







<b>assume </b><a NAME="E1:41"/>
<font color="Maroon">V</font> <a href="vectsp_5.html#R1">is_the_direct_sum_of</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font>
 ;<br/>

<a NAME="E2:41"/><b>then </b><i><font color="Green">E32</font></i>: 
(  <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">W2</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font> )
 
<b>by </b><i><a class="ref" href="vectsp_5.html#D4">VECTSP_5:def 4</a></i>;<br/>
<b>then </b> <a href="vectsp_4.html#K2">(Omega).</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">W2</font></span>)</span> = 
 <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font>
<b>by </b><i><a class="ref" href="vectsp_4.html#D4">VECTSP_4:def 4</a></i>
<br/>.= 
 <a href="vectsp_4.html#K1">(0).</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">W2</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_4.html#T47">VECTSP_4:47</a></i>
;<br/>
<a NAME="E4:41"/><b>then </b>
 <a href="vectsp_9.html#K1" target="_self">dim</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">W2</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="vectsp_9.html#E5"><i><font color="Green">Lemma30</font></i></a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="vectsp_9.html#K1" target="_self">dim</a> <font color="Maroon">W1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" target="_self">dim</a> <font color="Maroon">W2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="vectsp_9.html#K1" target="_self">dim</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">W2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0
<b>by </b><i/>
<br/>.= 
 <a href="vectsp_9.html#K1" target="_self">dim</a> <span class="p1">(<span class="default"><a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp_9.html#E2:41"><i><font color="Green">E32</font></i></a>, <a class="ref" href="vectsp_4.html#D4">VECTSP_4:def 4</a></i>
<br/>.= 
 <a href="vectsp_9.html#K1" target="_self">dim</a> <font color="Maroon">V</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E6:41"/>
 <a href="vectsp_9.html#K1" target="_self">dim</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" target="_self">dim</a> <font color="Maroon">W1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" target="_self">dim</a> <font color="Maroon">W2</font></span>)</span>
 ;<br/>


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