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<b>let </b><font color="Maroon" title="c1">F</font> be   <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">V</font> be  <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">VectSp</a> of <font color="Maroon" title="c1">F</font>;<br/><b>let </b><font color="Maroon" title="c3">W</font> be    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font>;<br/><b>let </b><font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c5">u</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font>;<br/>







<b>assume </b><a NAME="E1:17"/><i><font color="Green" title="E18">A1</font></i>: 
( not <font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W</font> &amp; not <font color="Maroon" title="c5">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W</font> &amp; <font color="Maroon" title="c4">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">u</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> &amp; <font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font> )
 ;<br/>

<a NAME="E2:17"/>
( <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E3:17"/><b>then </b>
( <font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <font color="Maroon" title="c5">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 
<b>by </b><i><a class="ref" href="vectsp_7.html#T13" title="VECTSP_7:th.13">VECTSP_7:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">v1</font> = <font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c7">u1</font> = <font color="Maroon" title="c5">u</font> as   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">L</font> = <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">u1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> as  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> <b>by </b><i><a class="txt" href="pencil_4.html#E1:17"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="vectsp_9.html#T16" title="VECTSP_9:th.16">VECTSP_9:16</a></i>;<br/>
<a NAME="E6:17"/>
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">u1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<b>by </b><i><a class="ref" href="vectsp_9.html#T21" title="VECTSP_9:th.21">VECTSP_9:21</a></i>;<br/>
<a NAME="E7:17"/><b>then </b>
 <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c8">L</font>
 
;<br/>
<a NAME="E8:17"/><b>then </b><i><font color="Green" title="E19">A2</font></i>: 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="txt" href="pencil_4.html#E1:17"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="vectsp_9.html#T35" title="VECTSP_9:th.35">VECTSP_9:35</a></i>;<br/>
<a NAME="E9:17"/>
<font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_4.html#E1:17"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="vectsp_4.html#T47" title="VECTSP_4:th.47">VECTSP_4:47</a></i>;<br/>
<a NAME="E10:17"/><b>then </b>
 <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span></span>)</span>
 
;<br/>
<a NAME="E11:17"/><b>then </b>
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="ref" href="vectsp_9.html#T33" title="VECTSP_9:th.33">VECTSP_9:33</a></i>;<br/>
<b>hence </b> <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p3">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p4"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p3">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p4"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">u</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c3">W</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2
<b>by </b><i><a class="txt" href="pencil_4.html#E8:17"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="vectsp_9.html#T36" title="VECTSP_9:th.36">VECTSP_9:36</a></i>
;<br/><br/>


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