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<b>let </b><font color="Maroon" title="c1">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>





<b>assume </b><a NAME="E1:80"/><i><font color="Green" title="E64">A1</font></i>: 
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p2</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="sppol_2.html#D1" target="_self" title="SPPOL_2:def.1">SPPOL_2:def 1</a><br/>

<b>given </b><font color="Maroon" title="c4">f1</font>, <font color="Maroon" title="c5">f2</font> being   <a href="sppol_2.html#NM1" title="SPPOL_2:NM.1">S-Sequence_in_R2</a><b> such that </b><a NAME="E2:80"/><i><font color="Green" title="E65">A2</font></i>: 
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 &amp; <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 )
 <b>and </b><br/><a NAME="E3:80"/><i><font color="Green" title="E66">A3</font></i>: 
( <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f1</font></span>)</span> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f2</font></span>)</span> )
 <b>and </b><br/><a NAME="E4:80"/><i><font color="Green" title="E67">A4</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>and </b><br/><a NAME="E5:80"/><i><font color="Green" title="E68">A5</font></i>: 
<font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c5">f2</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E6:80"/>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p1</font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E1:80"><i><font color="Green" title="E64">A1</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="sppol_2.html#D1" target="_self" title="SPPOL_2:def.1">SPPOL_2:def 1</a><br/>

<b>reconsider </b><font color="Maroon" title="c6">g1</font> =  <a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c4">f1</font>, <font color="Maroon" title="c7">g2</font> =  <a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c5">f2</font> as   <a href="sppol_2.html#NM1" title="SPPOL_2:NM.1">S-Sequence_in_R2</a> <b>by </b><i><a class="ref" href="sppol_2.html#T47" target="_self" title="SPPOL_2:th.47">Th47</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">g1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c7">g2</font>
;<br/>

<a NAME="E8:80"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f2</font> )
 
<b>by </b><i><a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></i>;<br/>
<b>hence </b><a NAME="E9:80"/>
( <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 &amp; <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span> &amp; <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="sppol_2.html#E2:80"><i><font color="Green" title="E65">A2</font></i></a>, <a class="txt" href="sppol_2.html#E3:80"><i><font color="Green" title="E66">A3</font></i></a>, <a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></i>;<br/>

<a NAME="E10:80"/>
(  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f1</font> &amp;  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c5">f2</font> )
 
<b>by </b><i><a class="ref" href="sppol_2.html#T22" target="_self" title="SPPOL_2:th.22">Th22</a></i>;<br/>
<b>hence </b><a NAME="E11:80"/>
( <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">g2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="sppol_2.html#E4:80"><i><font color="Green" title="E67">A4</font></i></a>, <a class="txt" href="sppol_2.html#E5:80"><i><font color="Green" title="E68">A5</font></i></a></i>;<br/>


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