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<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">q</font>, <font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">q1</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><b>that </b><br/><a NAME="E1:64"/><i><font color="Green" title="E42">A1</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font> is <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a>
 <b>and </b><br/><a NAME="E2:64"/><i><font color="Green" title="E43">A2</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">q1</font> is <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a>
 <b>and </b><br/><a NAME="E3:64"/><i><font color="Green" title="E44">A3</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>and </b><br/><a NAME="E4:64"/><i><font color="Green" title="E45">A4</font></i>: 
( <font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c3">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c4">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/>

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E5:64"/><i><font color="Green" title="E46">A5</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">q1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">r</font> = <font color="Maroon" title="c5">x</font> as   <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/>
<a NAME="E7:64"/><i><font color="Green" title="E47">A6</font></i>: 
<font color="Maroon" title="c3">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="txt" href="goboard7.html#E2:64"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="goboard7.html#E5:64"><i><font color="Green" title="E46">A5</font></i></a>, <a class="ref" href="sppol_1.html#T64" title="SPPOL_1:th.64">SPPOL_1:64</a></i>;<br/>
<a NAME="E8:64"/><i><font color="Green" title="E48">A7</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="txt" href="goboard7.html#E1:64"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="sppol_1.html#T37" title="SPPOL_1:th.37">SPPOL_1:37</a></i>;<br/>
<a NAME="E9:64"/>
( <font color="Maroon" title="c3">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c6">r</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">q1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="txt" href="goboard7.html#E4:64"><i><font color="Green" title="E45">A4</font></i></a>, <a class="txt" href="goboard7.html#E5:64"><i><font color="Green" title="E46">A5</font></i></a>, <a class="ref" href="topreal1.html#T10" title="TOPREAL1:th.10">TOPREAL1:10</a></i>;<br/>
<a NAME="E10:64"/><b>then </b>
( <font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c6">r</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="txt" href="goboard7.html#E4:64"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E11:64"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font>
 <b>by </b><i><a class="txt" href="goboard7.html#E3:64"><i><font color="Green" title="E44">A3</font></i></a>, <a class="txt" href="goboard7.html#E7:64"><i><font color="Green" title="E47">A6</font></i></a>, <a class="txt" href="goboard7.html#E8:64"><i><font color="Green" title="E48">A7</font></i></a>, <a class="ref" href="goboard7.html#T8" target="_self" title="GOBOARD7:th.8">Th8</a></i>;<br/>


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