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<b>let </b><font color="Maroon">n</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">f</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:9"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:9"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">j</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="E4:9"/><i><font color="Green">E43</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E5:9"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E6:9"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="sppol_2.html#T13">SPPOL_2:13</a></i>;<br/>
<a NAME="E7:9"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font>
 
<b>by </b><i>, , <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<b>take </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font>
;<br/>

<a NAME="E8:9"/>
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font>
 
<b>by </b><i>, , <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E9:9"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_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> <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="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<b>hence </b><a NAME="E10:9"/>
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="sppol_2.html#T5">SPPOL_2:5</a></i>;<br/>


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