<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</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">p1</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">p2</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">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="E1:82"/>
<font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="sppol_2.html#D1" target="_self">SPPOL_2:def 1</a><br/>

<b>given </b><font color="Maroon">f1</font> being   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a>, <font color="Maroon">f2</font> being   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a><b> such that </b><a NAME="E2:82"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1
 <b>and </b><br/><a NAME="E3:82"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> 1
 <b>and </b><br/><a NAME="E4:82"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span>
 <b>and </b><br/><a NAME="E5:82"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span>
 <b>and </b><br/><a NAME="E6:82"/><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span>
 <b>and </b><br/><a NAME="E7:82"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f2</font></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E8:82"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<b>assume </b><a NAME="E9:82"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p1</font>
 ;<br/>

<b>hence </b><a NAME="E10:82"/>
<font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="sppol_2.html#D1" target="_self">SPPOL_2:def 1</a><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:82_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font> or <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f2</font> )
 </span><b>by </b><i>, <a class="ref" href="sppol_2.html#T20" target="_self">Th20</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1"><b>suppose </b></a><a NAME="E1:82_1_1"/><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> <font color="Maroon">f1</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_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="E1:82_1_1_1_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font> or  not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:82_1_1_1_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:82_1_1_1_1_1"/>
 ex <font color="Olive">f1</font>, <font color="Olive">f2</font> being  <a href="sppol_2.html#NM1">S-Sequence_in_R2</a> st <br/>( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Olive">f1</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Olive">f2</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f1</font></span>)</span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f2</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">f2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">q</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">f1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">f2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="sppol_2.html#T14" target="_self">Th14</a>, <a class="ref" href="sppol_2.html#T15" target="_self">Th15</a>, <a class="ref" href="sppol_2.html#T18" target="_self">Th18</a>, , <a class="ref" href="sppol_2.html#T19" target="_self">Th19</a>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:82_1_1_1_1_2"/><i><font color="Green">E51</font></i>: 
not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">i</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="E3:82_1_1_1_1_2"/><i><font color="Green">E52</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E4:82_1_1_1_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">i</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">f1</font>
 <b>and </b><br/><a NAME="E5:82_1_1_1_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:82_1_1_1_1_2"><i><font color="Green">E74</font></i></a>, </i>;<br/><b>reconsider </b><font color="Maroon">f1'</font> =  <a href="finseq_5.html#K5">Ins</a> <font color="Maroon">f1</font>,<font color="Maroon">i</font>,<font color="Maroon">q</font> as   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a> <b>by </b><i><a class="txt" href="sppol_2.html#E5:82_1_1_1_1_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="sppol_2.html#E10:82_1_1_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="sppol_2.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E7:82_1_1_1_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1'</font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T74">FINSEQ_5:74</a></i>;<br/><a NAME="E8:82_1_1_1_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1'</font> <a href="finseq_4.html#K4">/.</a> 1
 <b>by </b><i><a class="ref" href="sppol_2.html#T14" target="_self">Th14</a>, <a class="txt" href="sppol_2.html#E7:82_1_1_1_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="finseq_5.html#T78">FINSEQ_5:78</a></i>;<br/><a NAME="E9:82_1_1_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_5.html#T72">FINSEQ_5:72</a></i>;<br/><a NAME="E10:82_1_1_1_1_2"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1'</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1'</font></span>)</span>
 <b>by </b><i><a class="ref" href="sppol_2.html#T18" target="_self">Th18</a>, <a class="txt" href="sppol_2.html#E8:82_1_1_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="finseq_5.html#T77">FINSEQ_5:77</a></i>;<br/><a NAME="E11:82_1_1_1_1_2"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">f1'</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E10:82_1_1_1_1_2"><i><font color="Green">E78</font></i></a>, </i>;<br/><b>hence </b><a NAME="E12:82_1_1_1_1_2"/>
 ex <font color="Olive">f1</font>, <font color="Olive">f2</font> being  <a href="sppol_2.html#NM1">S-Sequence_in_R2</a> st <br/>( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Olive">f1</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Olive">f2</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f1</font></span>)</span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f2</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">f2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">q</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">f1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">f2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="sppol_2.html#T15" target="_self">Th15</a>, , <a class="ref" href="sppol_2.html#T19" target="_self">Th19</a>, , , , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:82_1_1"/>
 ex <font color="Olive">f1</font>, <font color="Olive">f2</font> being  <a href="sppol_2.html#NM1">S-Sequence_in_R2</a> st <br/>( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Olive">f1</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Olive">f2</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f1</font></span>)</span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f2</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">f2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">q</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">f1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">f2</font></span>)</span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_2"><b>suppose </b></a><a NAME="E1:82_1_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:82_1_2_1_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font> or  not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:82_1_2_1_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:82_1_2_1_1_1"/>
 ex <font color="Olive">f1</font>, <font color="Olive">f2</font> being  <a href="sppol_2.html#NM1">S-Sequence_in_R2</a> st <br/>( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Olive">f1</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Olive">f2</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f1</font></span>)</span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f2</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">f2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">q</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">f1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">f2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="sppol_2.html#T14" target="_self">Th14</a>, <a class="ref" href="sppol_2.html#T15" target="_self">Th15</a>, <a class="ref" href="sppol_2.html#T18" target="_self">Th18</a>, , <a class="ref" href="sppol_2.html#T19" target="_self">Th19</a>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:82_1_2_1_1_2"/><i><font color="Green">E80</font></i>: 
not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">i</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="E3:82_1_2_1_1_2"/><i><font color="Green">E81</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E4:82_1_2_1_1_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">i</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">f2</font>
 <b>and </b><br/><a NAME="E5:82_1_2_1_1_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f2</font>,<font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">f2'</font> =  <a href="finseq_5.html#K5">Ins</a> <font color="Maroon">f2</font>,<font color="Maroon">i</font>,<font color="Maroon">q</font> as   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a> <b>by </b><i>, , <a class="ref" href="sppol_2.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E7:82_1_2_1_1_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f2'</font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T74">FINSEQ_5:74</a></i>;<br/><a NAME="E8:82_1_2_1_1_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2'</font> <a href="finseq_4.html#K4">/.</a> 1
 <b>by </b><i><a class="ref" href="sppol_2.html#T15" target="_self">Th15</a>, , <a class="ref" href="finseq_5.html#T78">FINSEQ_5:78</a></i>;<br/><a NAME="E9:82_1_2_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_5.html#T72">FINSEQ_5:72</a></i>;<br/><a NAME="E10:82_1_2_1_1_2"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2'</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2'</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_5.html#T77">FINSEQ_5:77</a></i>;<br/><a NAME="E11:82_1_2_1_1_2"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">f2'</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f2</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E12:82_1_2_1_1_2"/>
 ex <font color="Olive">f1</font>, <font color="Olive">f2</font> being  <a href="sppol_2.html#NM1">S-Sequence_in_R2</a> st <br/>( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Olive">f1</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Olive">f2</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f1</font></span>)</span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f2</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">f2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">q</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">f1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">f2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="sppol_2.html#T14" target="_self">Th14</a>, <a class="ref" href="sppol_2.html#T18" target="_self">Th18</a>, <a class="ref" href="sppol_2.html#T19" target="_self">Th19</a>, , , <a class="txt" href="sppol_2.html#E1:82_1_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="sppol_2.html#E1:82_1_2_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="sppol_2.html#E3:82_1_2_1_1_2"><i><font color="Green">E81</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:82_1_2"/>
 ex <font color="Olive">f1</font>, <font color="Olive">f2</font> being  <a href="sppol_2.html#NM1">S-Sequence_in_R2</a> st <br/>( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Olive">f1</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Olive">f2</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f1</font></span>)</span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f2</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">f2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">q</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">f1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">f2</font></span>)</span> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
