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<b>let </b><font color="Maroon">V</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">I</font> be    <a href="rusub_3.html#M1" target="_self">Basis</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/>





<b>assume </b><a NAME="E1:34"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">I</font>
 ;<br/>

<b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/>

<b>assume </b><a NAME="E2:34"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A</font>
 ;<br/>

<b>assume </b><a NAME="E3:34"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">B</font> is <a href="rlvect_3.html#V1">linearly-independent</a>
 ;<br/>

<b>consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:34"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">Bv</font> = <font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> ;<br/>
<a NAME="E8:34"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="txt" href="rusub_3.html#E1:34"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E9:34"/><b>then </b><i><font color="Green">E33</font></i>: 
not <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="rusub_3.html#K1" target="_self">Lin</a> <font color="Maroon">Bv</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E10:34"/>
<font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="txt" href="rusub_3.html#E1:34"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E11:34"/><b>then </b>
( <font color="Maroon">I</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> &amp; not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a>, <a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/>
<a NAME="E12:34"/><b>then </b>
<font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Bv</font>
 
<b>by </b><i><a class="ref" href="rusub_3.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E13:34"/><b>then </b>
(  <a href="rusub_3.html#K1" target="_self">Lin</a> <font color="Maroon">I</font> is    <a href="rusub_1.html#M1">Subspace</a> of  <a href="rusub_3.html#K1" target="_self">Lin</a> <font color="Maroon">Bv</font> &amp; <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="rusub_3.html#K1" target="_self">Lin</a> <font color="Maroon">I</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E14:34"/>
contradiction
 <b>by </b><i>, <a class="ref" href="rusub_1.html#T1">RUSUB_1:1</a></i>;<br/>


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