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<b>let </b><font color="Maroon">V</font> be  <a href="rusub_4.html#V1" target="_self">finite-dimensional</a> <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:21_1"/><i><font color="Green">E26</font></i>: 
 <a href="rusub_4.html#K1" target="_self">dim</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>consider </b><font color="Maroon">I</font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:21_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">I</font> is    <a href="rusub_3.html#M1">Basis</a> of <font color="Maroon">V</font>
 <b>by </b><i/>;<br/><a NAME="E4:21_1"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , </i>;<br/><a NAME="E5:21_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/> <a href="rusub_1.html#K2">(Omega).</a> <font color="Maroon">V</font> = 
 <a href="bhsp_1.html#G1">UNITSTR</a>(# the <a href="struct_0.html#U1">carrier</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="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font>,the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">V</font> #)
<b>by </b><i><a class="ref" href="rusub_1.html#D3">RUSUB_1:def 3</a></i>
<br/>.= 
 <a href="rusub_3.html#K1">Lin</a> <font color="Maroon">I</font>
<b>by </b><i>, <a class="ref" href="rusub_3.html#D2">RUSUB_3:def 2</a></i>
<br/>.= 
 <a href="rusub_1.html#K1">(0).</a> <font color="Maroon">V</font>
<b>by </b><i>, <a class="ref" href="rusub_3.html#T3">RUSUB_3:3</a></i>
;<br/><b>hence </b><a NAME="E7:21_1"/>
 <a href="rusub_1.html#K2">(Omega).</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="rusub_1.html#K1">(0).</a> <font color="Maroon">V</font>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:21"/>
 <a href="rusub_1.html#K2">(Omega).</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="rusub_1.html#K1">(0).</a> <font color="Maroon">V</font>
 ;<br/>

<a NAME="E3:21"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="bhsp_1.html#G1">UNITSTR</a>(# the <a href="struct_0.html#U1">carrier</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="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font>,the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">V</font> #) <a href="hidden.html#R1">=</a>  <a href="rusub_1.html#K1">(0).</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="rusub_1.html#D3">RUSUB_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">I</font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E5:21"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">I</font> is    <a href="rusub_3.html#M1">Basis</a> of <font color="Maroon">V</font>
 <b>by </b><i/>;<br/>
<a NAME="E6:21"/>
 <a href="rusub_3.html#K1">Lin</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="rusub_1.html#K1">(0).</a> <font color="Maroon">V</font>
 
<b>by </b><i>, , <a class="ref" href="rusub_3.html#D2">RUSUB_3:def 2</a></i>;<br/>
<a NAME="E7:21"/><b>then </b><i><font color="Green">E33</font></i>: 
( <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="rusub_3.html#T4">RUSUB_3:4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_2"/>
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E2:21_2"/><b>then </b>
not <font color="Maroon">I</font> is <a href="rlvect_3.html#V1">linearly-independent</a>
 <b>by </b><i><a class="ref" href="rlvect_3.html#T9">RLVECT_3:9</a></i>;<br/><b>hence </b><a NAME="E3:21_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="rusub_3.html#D2">RUSUB_3:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:21"/>
 <a href="rusub_4.html#K1" target="_self">dim</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , , <a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/>


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