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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/>







<i><font color="Green">E32</font></i>: <span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><a href="vectsp_9.html#K1" target="_self">dim</a> <font color="Maroon">W2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" target="_self">dim</a> <font color="Maroon">V</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_9.html#K1" target="_self">dim</a> <span class="p3">(<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> <span class="p2">(<span class="default"><a href="vectsp_9.html#K1" target="_self">dim</a> <span class="p3">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">W2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" target="_self">dim</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_9.html#K1" target="_self">dim</a> <span class="p3">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">W2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="vectsp_9.html#K1" target="_self">dim</a> <font color="Maroon">V</font></span>)</span></span>)</span>

;<br/>
<a NAME="E2:40"/><i><font color="Green">E40</font></i>: 
 <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#K1">+</a> <font color="Maroon">W2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="vectsp_9.html#K1" target="_self">dim</a> <font color="Maroon">V</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:40"/>
<span class="p1">(<span class="default"><a href="vectsp_9.html#K1" target="_self">dim</a> <font color="Maroon">V</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_9.html#K1" target="_self">dim</a> <span class="p3">(<span class="default"><font color="Maroon">W1</font> <a href="vectsp_5.html#K2">/\</a> <font color="Maroon">W2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="vectsp_9.html#K1" target="_self">dim</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <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>
 
;<br/>
<b>hence </b><a NAME="E4:40"/>
 <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="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><a href="vectsp_9.html#K1" target="_self">dim</a> <font color="Maroon">W2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" target="_self">dim</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i><a class="txt" href="vectsp_9.html#E1:40"><i><font color="Green">E32</font></i></a>, <a class="ref" href="vectsp_9.html#T5" target="_self">Th5</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>


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