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<b>let </b><font color="Maroon">j</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">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:65"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:65"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K2" target="_self">:-</a> <font color="Maroon">p</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">i</font> = <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>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="E4:65"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E5:65"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">f</font> <a href="finseq_5.html#K2" target="_self">:-</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">k</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E6:65"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font> &amp; <font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_5.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_4.html#T31">FINSEQ_4:31</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E7:65"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:65"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="txt" href="finseq_5.html#E8:65"><i><font color="Green">E36</font></i></a>, </i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K2" target="_self">:-</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_5.html#K2" target="_self">:-</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="finseq_5.html#E8:65"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="txt" href="finseq_5.html#E8:65"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_5.html#T8" target="_self">Th8</a>, , <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#T6" target="_self">Th6</a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>


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