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<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</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="E1:24"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span> )
 ;<br/>

<a NAME="E2:24"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">q</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">q</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E3:24"/><b>then </b><i><font color="Green">E35</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font> &amp; <font color="Maroon">j</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="relat_1.html#T20">RELAT_1:20</a></i>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:24"/><i><font color="Green">E39</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">k</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/>
<a NAME="E6:24"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:24"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E8:24"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="funct_4.html#T71">FUNCT_4:71</a></i>;<br/>
<a NAME="E9:24"/><i><font color="Green">E67</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="E10:24"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T24">RELAT_1:24</a></i>;<br/>
<b>then </b> <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q</font> = 
<font color="Maroon">q</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
<b>by </b><i>, , , <a class="ref" href="finseq_3.html#T51">FINSEQ_3:51</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
<b>by </b><i>, <a class="ref" href="finseq_2.html#T145">FINSEQ_2:145</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>
;<br/>
<b>hence </b><a NAME="E12:24"/>
 <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#K10">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 ;<br/>


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