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<b>let </b><font color="Maroon">R</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_2.html#NM6">LeftMod</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:4"/><i><font color="Green">E26</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E2:4"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">A</font> is <a href="lmod_5.html#V1" target="_self">linearly-independent</a>
 <b>and </b><br/><a NAME="E3:4"/><i><font color="Green">E28</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> =  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E5:4"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E6:4"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>  st <font color="Olive">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> holds <br/><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">v</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S6">FUNCT_2:sch 6</a>();<br/></i>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<a NAME="E8:4"/>
 ex <font color="Olive">T</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> st <br/>for <font color="Olive">v</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>  st not <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Olive">T</font> holds <br/><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">T</font> = <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/>

<b>let </b><font color="Maroon">v</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/>

<b>assume </b><a NAME="E1:4_1"/>
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 ;<br/>

<a NAME="E2:4_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:4_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="vectsp_6.html#M1">Linear_Combination</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="vectsp_6.html#D4">VECTSP_6:def 4</a></i>;<br/>
<a NAME="E10:4"/><i><font color="Green">E36</font></i>: 
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">f</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:4_2"/>
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</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>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:4_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:4_2_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E4:4_2_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 ;<br/>
<a NAME="E5:4_2_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:4_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:4_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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="E3:4_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> &amp;  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:4_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font> : <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:4_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">f</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:4"/><b>then </b>
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/>
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp_6.html#T46">VECTSP_6:46</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T59">VECTSP_1:59</a></i>
;<br/>
<b>hence </b><a NAME="E14:4"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div>
