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<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">q</font> be   <a href="finseq_1.html#NM4">FinSubsequence</a>;<br/>



<b>assume </b><a NAME="E1:6"/>
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span>
 ;<br/>

<a NAME="E2:6"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D5">FINSEQ_1:def 5</a></i>;<br/>
<a NAME="E3:6"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T23">RELAT_1:23</a></i>;<br/>
<a NAME="E4:6"/><i><font color="Green">E35</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T23">RELAT_1:23</a></i>;<br/>
<a NAME="E5:6"/><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>;<br/>
<a NAME="E6:6"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T71">FINSEQ_1:71</a></i>;<br/>
<a NAME="E7:6"/><b>then </b><i><font color="Green">E38</font></i>: 
( <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> )
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#T46">RELAT_1:46</a>, <a class="ref" href="relat_1.html#T47">RELAT_1:47</a></i>;<br/>
<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:6"/><i><font color="Green">E41</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/>
<a NAME="E10:6"/>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T45">FINSEQ_3:45</a></i>;<br/>
<a NAME="E11:6"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:6"/><i><font color="Green">E43</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="card_1.html#T49">CARD_1:49</a></i>;<br/>
<a NAME="E14:6"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E15:6"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E17:6"/>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, , , <a class="ref" href="pre_circ.html#T2">PRE_CIRC:2</a></i>;<br/>
<b>hence </b><a NAME="E18:6"/>
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>


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