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<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">p</font> be   <a href="finseq_1.html#NM4">FinSubsequence</a>, <font color="Maroon">q</font> be   <a href="finseq_1.html#NM4">FinSubsequence</a>;<br/>



<b>assume </b><a NAME="E1:106"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:106"/><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">k</font></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</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="E3:106"/><i><font color="Green">E35</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">p</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">k</font></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</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/>
<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="relat_1.html#D3">RELAT_1:def 3</a><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E4:106"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i</font> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E5:106"/><b>then </b><i><font color="Green">E37</font></i>: 
( <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> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</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="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">k</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="E7:106"/><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">k</font> &amp; <font color="Maroon">k</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="E8:106"/><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="relat_1.html#K1">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/>
<a NAME="E9:106"/><b>then </b><i><font color="Green">E43</font></i>: 
<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">p</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<font color="Maroon">y</font> = 
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, , <a class="txt" href="pnproc_1.html#E10"><i><font color="Green">Lemma50</font></i></a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="txt" href="pnproc_1.html#E10"><i><font color="Green">Lemma50</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E11:106"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">i</font> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>


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