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<div class="add">

<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>assume </b><a NAME="E1:53"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">f</font> is <a href="topreal1.html#V1">special</a> &amp; <font color="Maroon">f</font> is <a href="jordan23.html#V2" target="_self">weakly-one-to-one</a> )
 ;<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>, <font color="Maroon">q</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>assume </b><a NAME="E2:53"/><i><font color="Green">E45</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> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> )
 ;<br/>

<a NAME="E3:53"/><i><font color="Green">E46</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> &amp;  <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<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="jordan3.html#T41">JORDAN3:41</a></i>;<br/>
<a NAME="E4:53"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <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="hsNdiv(this)" href="javascript:()" title="53_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> or <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1"><b>suppose </b></a><a NAME="E1:53_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1_1_1_1"/>
( ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> &amp;  <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> ) or (  <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> &amp;  <a href="jordan3.html#R2">LE</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span>,<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">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( not <!--HUMANRECFAILED-->( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> &amp;  <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> &amp;  <a href="jordan3.html#R2">LE</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span>,<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">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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:53_1_1_1_1_1"/><i><font color="Green">E49</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> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> &amp;  <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> )
 ;<br/><div class="add"><a NAME="E2:53_1_1_1_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan3.html#T64">JORDAN3:64</a></i>;<br/><a NAME="E3:53_1_1_1_1_1"/>
 <a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font> is <a href="topreal1.html#V1">special</a>
 <b>by </b><i>, , </i>;<br/><a NAME="E4:53_1_1_1_1_1"/><b>then </b>
 <a href="jordan3.html#K4">R_Cut</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">q</font> is <a href="topreal1.html#V1">special</a>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:53_1_1_1_1_1"/>
 <a href="jordan3.html#K5">B_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font>,<font color="Maroon">q</font> is <a href="topreal1.html#V1">special</a>
 <b>by </b><i><a class="ref" href="jordan23.html#T1" target="_self">Th1</a>, <a class="ref" href="jordan3.html#D8">JORDAN3:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:53_1_1_1_1_2"/><i><font color="Green">E51</font></i>: 
(  <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> &amp;  <a href="jordan3.html#R2">LE</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span>,<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">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/><div class="add"><a NAME="E2:53_1_1_1_1_2"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan3.html#T66">JORDAN3:66</a></i>;<br/><a NAME="E3:53_1_1_1_1_2"/>
 <a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font> is <a href="topreal1.html#V1">special</a>
 <b>by </b><i>, , </i>;<br/><a NAME="E4:53_1_1_1_1_2"/><b>then </b>
 <a href="jordan3.html#K4">R_Cut</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">q</font> is <a href="topreal1.html#V1">special</a>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:53_1_1_1_1_2"/>
 <a href="jordan3.html#K5">B_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font>,<font color="Maroon">q</font> is <a href="topreal1.html#V1">special</a>
 <b>by </b><i><a class="ref" href="jordan23.html#T1" target="_self">Th1</a>, <a class="ref" href="jordan3.html#D8">JORDAN3:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:53_1_1_1_1_3"/><i><font color="Green">E53</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> &amp;  <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> &amp;  <a href="jordan3.html#R2">LE</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span>,<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">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) )
 ;<br/><div class="add"><a NAME="E2:53_1_1_1_1_3"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="jordan3.html#K5">B_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><a href="jordan3.html#K4">R_Cut</a> <span class="p2">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">q</font></span>)</span>,<font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan3.html#D8">JORDAN3:def 8</a></i>;<br/><div><i><font color="Green">E62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1_1_1_1_3_1_1"/>
(  <a href="jordan3.html#K2">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> or  <a href="jordan3.html#K2">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> )
 </span><b>by </b><i>, <a class="ref" href="jordan23.html#T1" target="_self">Th1</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:53_1_1_1_1_3_1_1_1"/>
 <a href="jordan3.html#K2">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:53_1_1_1_1_3_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan3.html#T64">JORDAN3:64</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:53_1_1_1_1_3_1_1_2"/><i><font color="Green">E63</font></i>: 
 <a href="jordan3.html#K2">Index</a> <font color="Maroon">q</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">Ls</font> =  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">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>;<br/><a NAME="E2:53_1_1_1_1_3_1_1_2"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K2">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="E3:53_1_1_1_1_3_1_1_2"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K2">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#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E4:53_1_1_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K2">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="jordan23.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:53_1_1_1_1_3_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">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>, , </i>;<br/><a NAME="E6:53_1_1_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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">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="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E7:53_1_1_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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">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="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E8:53_1_1_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">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> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span>,<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">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="topreal1.html#T15">TOPREAL1:15</a></i>;<br/><a NAME="E9:53_1_1_1_1_3_1_1_2"/><i><font color="Green">E68</font></i>: 
not  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>, <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">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>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span>,<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">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="jordan23.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan23.html#E1:53"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="jordan5c.html#T17">JORDAN5C:17</a></i>;<br/><a NAME="E10:53_1_1_1_1_3_1_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">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>
 <b>by </b><i>, <a class="ref" href="jordan5b.html#T32">JORDAN5B:32</a></i>;<br/><a NAME="E11:53_1_1_1_1_3_1_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">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>
 <b>by </b><i>, <a class="txt" href="jordan23.html#E1:53"><i><font color="Green">E35</font></i></a>, <a class="ref" href="jordan5b.html#T32">JORDAN5B:32</a></i>;<br/><a NAME="E12:53_1_1_1_1_3_1_1_2"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q</font>,<font color="Maroon">p</font>, <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">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>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span>,<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">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="jordan23.html#D1" target="_self">Def1</a>, , , <a class="ref" href="jordan5c.html#T14">JORDAN5C:14</a></i>;<br/><a NAME="E13:53_1_1_1_1_3_1_1_2"/><b>then </b>
 <a href="jordan3.html#R2">LE</a> <font color="Maroon">q</font>,<font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span>,<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">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="jordan5c.html#T17">JORDAN5C:17</a></i>;<br/><b>hence </b><a NAME="E14:53_1_1_1_1_3_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="jordan23.html#E1:53"><i><font color="Green">E35</font></i></a>, <a class="ref" href="jordan3.html#T66">JORDAN3:66</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E4:53_1_1_1_1_3"/>
 <a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">q</font> is <a href="topreal1.html#V1">special</a>
 <b>by </b><i>, , </i>;<br/><a NAME="E5:53_1_1_1_1_3"/><b>then </b>
 <a href="jordan3.html#K4">R_Cut</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">q</font></span>)</span>,<font color="Maroon">p</font> is <a href="topreal1.html#V1">special</a>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E6:53_1_1_1_1_3"/>
 <a href="jordan3.html#K5">B_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font>,<font color="Maroon">q</font> is <a href="topreal1.html#V1">special</a>
 <b>by </b><i>, <a class="ref" href="sppol_2.html#T42">SPPOL_2:42</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:53_1_1"/>
 <a href="jordan3.html#K5">B_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font>,<font color="Maroon">q</font> is <a href="topreal1.html#V1">special</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_2"><b>suppose </b></a><a NAME="E1:53_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/><div class="add"><a NAME="E2:53_1_2"/><b>then </b>
 <a href="jordan3.html#K5">B_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:53_1_2"/>
 <a href="jordan3.html#K5">B_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font>,<font color="Maroon">q</font> is <a href="topreal1.html#V1">special</a>
 <b>by </b><i><a class="ref" href="sppol_2.html#T39">SPPOL_2:39</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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