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<b>let </b><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">q</font> be   <a href="finseq_1.html#NM4" title="FINSEQ_1:NM.4">FinSubsequence</a>;<br/>



<b>assume </b><a NAME="E1:6"/>
 <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 ;<br/>

<a NAME="E2:6"/><b>then </b><i><font color="Green" title="E6">A1</font></i>: 
 <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<font color="Maroon" title="c1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D5" title="FINSEQ_1:def.5">FINSEQ_1:def 5</a></i>;<br/>
<a NAME="E3:6"/><b>then </b><i><font color="Green" title="E7">A2</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T23" title="RELAT_1:th.23">RELAT_1:23</a></i>;<br/>
<a NAME="E4:6"/><i><font color="Green" title="E8">A3</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E2:6"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="relat_1.html#T23" title="RELAT_1:th.23">RELAT_1:23</a></i>;<br/>
<a NAME="E5:6"/><i><font color="Green" title="E9">A4</font></i>: 
 <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">q</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D14" title="FINSEQ_1:def.14">FINSEQ_1:def 14</a></i>;<br/>
<a NAME="E6:6"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">q</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T71" title="FINSEQ_1:th.71">FINSEQ_1:71</a></i>;<br/>
<a NAME="E7:6"/><b>then </b><i><font color="Green" title="E10">A5</font></i>: 
( <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">q</font> )
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E3:6"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="pnproc_1.html#E5:6"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a>, <a class="ref" href="relat_1.html#T47" title="RELAT_1:th.47">RELAT_1:47</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E9:6"/><i><font color="Green" title="E11">A6</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D12" title="FINSEQ_1:def.12">FINSEQ_1:def 12</a></i>;<br/>
<a NAME="E10:6"/>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E7:6"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="pnproc_1.html#E9:6"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="finseq_3.html#T45" title="FINSEQ_3:th.45">FINSEQ_3:45</a></i>;<br/>
<a NAME="E11:6"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T78" title="FINSEQ_1:th.78">FINSEQ_1:78</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:6"/><i><font color="Green" title="E12">A7</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="card_1.html#T49" title="CARD_1:th.49">CARD_1:49</a></i>;<br/>
<a NAME="E14:6"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">q</font>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E13:6"><i><font color="Green" title="E12">A7</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E15:6"/><b>then </b>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E9:6"><i><font color="Green" title="E11">A6</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c4">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E17:6"/>
<font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E4:6"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="pnproc_1.html#E7:6"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="pnproc_1.html#E13:6"><i><font color="Green" title="E12">A7</font></i></a>, <a class="ref" href="pre_circ.html#T2" title="PRE_CIRC:th.2">PRE_CIRC:2</a></i>;<br/>
<b>hence </b><a NAME="E18:6"/>
 ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>


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