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<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">p</font> be   <a href="pre_topc.html#NM2">Point</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">i1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:33"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">f</font> is <a href="topreal1.html#V2">unfolded</a> &amp; <font color="Maroon">f</font> is <a href="topreal1.html#V3">s.n.c.</a> )
 <b>and </b><br/><a NAME="E2:33"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">i1</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> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E3:33"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>
 ;<br/>

<b>assume </b><a NAME="E4:33"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font>
 ;<br/>

<a NAME="E5:33"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="sppol_2.html#T17">SPPOL_2:17</a></i>;<br/>
<a NAME="E6:33"/><i><font color="Green">E62</font></i>: 
 <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font>
 
<b>by </b><i>, <a class="ref" href="jordan3.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E7:33"/>
<font color="Maroon">p</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"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan3.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E8:33"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:33"/><i><font color="Green">E64</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="jordan3.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E10:33"/><i><font color="Green">E65</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E11:33"/>
<font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E12:33"/><b>then </b>
 <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="xxreal_0.html#NR3" target="_self">&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="E13:33"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</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>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E14:33"/><b>then </b><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <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="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E2:33_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <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/>;<br/><a NAME="E3:33_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="jordan3.html#T13" target="_self">Th13</a>, , , , <a class="ref" href="topreal1.html#D8">TOPREAL1:def 8</a></i>;<br/><a NAME="E4:33_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:33_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jordan3.html#T15" target="_self">Th15</a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E16:33"/>
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="jordan3.html#T13" target="_self">Th13</a>, , <a class="ref" href="jordan3.html#T11" target="_self">Th11</a></i>;<br/>


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