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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>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="E2:23_1"/><i><font color="Green">E26</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/><b>assume </b><a NAME="E3:23_1"/>
 <a href="rusub_4.html#K1" target="_self">dim</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><a NAME="E4:23_1"/><b>then </b><i><font color="Green">E27</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:23_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="card_1.html#T47">CARD_1:47</a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1"/>
<font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E2:23_1_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">I</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/><a NAME="E3:23_1_1"/><b>then </b>
2 <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i>, <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>;<br/><b>hence </b><a NAME="E4:23_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:23_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</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/><a NAME="E12:23_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">u</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E13:23_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="domain_1.html#K7">}</a></span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E14:23_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="domain_1.html#K7">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_2"/>
not <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="domain_1.html#K7">}</a></span>
 ;<br/><b>then consider </b><font color="Maroon">w</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:23_1_2"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; not <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="domain_1.html#K7">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E4:23_1_2"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">u</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E5:23_1_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">w</font></span><a href="enumset1.html#K1">}</a></span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, , , <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><a NAME="E6:23_1_2"/><b>then </b>
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">w</font></span><a href="enumset1.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E7:23_1_2"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">w</font></span><a href="enumset1.html#K1">}</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/><a NAME="E8:23_1_2"/><b>then </b>
3 <a href="xxreal_0.html#R1">&lt;=</a> 2
 <b>by </b><i>, <a class="ref" href="rusub_4.html#T1" target="_self">Th1</a>, , <a class="ref" href="card_2.html#T77">CARD_2:77</a></i>;<br/><b>hence </b><a NAME="E9:23_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E16:23_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>then </b><i><font color="Green">E40</font></i>:  <a href="rusub_3.html#K1">Lin</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="domain_1.html#K7">}</a></span> = 
 <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_3.html#D2">RUSUB_3:def 2</a></i>
<br/>.= 
 <a href="rusub_1.html#K2">(Omega).</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/><a NAME="E18:23_1"/>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="domain_1.html#K7">}</a></span> is <a href="rlvect_3.html#V1">linearly-independent</a>
 <b>by </b><i>, , <a class="ref" href="rusub_3.html#D2">RUSUB_3:def 2</a></i>;<br/><b>hence </b><a NAME="E19:23_1"/>
 ex <font color="Olive">u</font>, <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">v</font> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">u</font>,<font color="Olive">v</font></span><a href="domain_1.html#K7">}</a></span> is <a href="rlvect_3.html#V1">linearly-independent</a> &amp;  <a href="rusub_1.html#K2">(Omega).</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="rusub_3.html#K1">Lin</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">u</font>,<font color="Olive">v</font></span><a href="domain_1.html#K7">}</a></span> )
 <b>by </b><i><a class="ref" href="rusub_4.html#T1" target="_self">Th1</a>, </i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">u</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">v</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><a NAME="E2:23"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E3:23"/><i><font color="Green">E42</font></i>: 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="domain_1.html#K7">}</a></span> is <a href="rlvect_3.html#V1">linearly-independent</a>
 <b>and </b><br/><a NAME="E4:23"/><i><font color="Green">E43</font></i>: 
 <a href="rusub_1.html#K2">(Omega).</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="rusub_3.html#K1">Lin</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="domain_1.html#K7">}</a></span>
 ;<br/>

<a NAME="E5:23"/>
 <a href="rusub_3.html#K1">Lin</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a>  <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 NAME="E6:23"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="domain_1.html#K7">}</a></span> is    <a href="rusub_3.html#M1">Basis</a> of <font color="Maroon">V</font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> 2 )
 
<b>by </b><i>, , <a class="ref" href="card_2.html#T76">CARD_2:76</a>, <a class="ref" href="rusub_3.html#D2">RUSUB_3:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:23"/>
 <a href="rusub_4.html#K1" target="_self">dim</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i/>;<br/>


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