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<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</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="E1:114"/><i><font color="Green">E29</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>
 ;<br/>

<a NAME="E2:114"/><i><font color="Green">E30</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E10"><i><font color="Green">Lemma50</font></i></a></i>;<br/>
<a NAME="E3:114"/><i><font color="Green">E35</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/>
<a NAME="E4:114"/><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/>
<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="E5:114"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">j1</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="E7:114"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">j1</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/>;<br/>
<a NAME="E8:114"/>
<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="pnproc_1.html#T13" target="_self">Th13</a></i>;<br/>
<b>then consider </b><font color="Maroon">j2</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="E10:114"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="pnproc_1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E11:114"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E12:114"/><b>then </b><i><font color="Green">E43</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font>
 
<b>by </b><i><a class="ref" href="pnproc_1.html#T14" target="_self">Th14</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E13:114"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="pnproc_1.html#T14" target="_self">Th14</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E14:114"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="pnproc_1.html#E22"><i><font color="Green">Lemma80</font></i></a>, <a class="ref" href="pnproc_1.html#T13" target="_self">Th13</a>, <a class="ref" href="pnproc_1.html#D5" target="_self">Def5</a></i>;<br/>


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