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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">F<sub>1</sub></font>()) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  = 0;<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">F<sub>1</sub></font>(), <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E2:4_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>() &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>() &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">F<sub>4</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>() )
 <b>and </b><br/><a NAME="E3:4_1"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">u</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Olive">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Olive">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F<sub>4</sub></font>() holds <br/><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">u</font>)
 <b>from </b><i><a class="ref" href="rlvect_4.html#S1" target="_self">RLVECT_4:sch 1</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">F<sub>1</sub></font>(),<a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>assume </b><a NAME="E1:4_1_1"/>
not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="domain_1.html#K8">}</a></span>
 ;<br/><a NAME="E2:4_1_1"/><b>then </b>
( <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F<sub>4</sub></font>() )
 <b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:4_1_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 0
 <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="rlvect_2.html#M1">Linear_Combination</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="rlvect_2.html#D5">RLVECT_2:def 5</a></i>;<br/>
<a NAME="E7:4_1"/>
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="domain_1.html#K8">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2"><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><b>that </b><br/><a NAME="E1:4_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:4_1_2"/><i><font color="Green">E63</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="domain_1.html#K8">}</a></span>
 ;<br/>

<a NAME="E3:4_1_2"/>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F<sub>4</sub></font>() )
 
<b>by </b><i>, , <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a>, <a class="ref" href="rlvect_2.html#T28">RLVECT_2:28</a></i>;<br/>
<b>hence </b><a NAME="E4:4_1_2"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="rlvect_2.html#M2">Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="domain_1.html#K8">}</a></span> <b>by </b><i><a class="ref" href="rlvect_2.html#D8">RLVECT_2:def 8</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E9:4_1"/>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>() &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>() &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">F<sub>4</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>() )
 <b>by </b><i/>;<br/>


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