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<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>, <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>, <font color="Maroon">q1</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><b>that </b><br/><a NAME="E1:66"/><i><font color="Green">E43</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> is <a href="sppol_1.html#V1">horizontal</a>
 <b>and </b><br/><a NAME="E2:66"/><i><font color="Green">E47</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">q1</font> is <a href="sppol_1.html#V1">horizontal</a>
 <b>and </b><br/><a NAME="E3:66"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a> 
 <b>and </b><br/><a NAME="E4:66"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  )
 ;<br/>

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

<b>assume </b><a NAME="E5:66"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">q1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">r</font> = <font color="Maroon">x</font> as   <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/>
<a NAME="E7:66"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="ref" href="goboard7.html#T2" target="_self">Th2</a>, <a class="txt" href="goboard7.html#E2:66"><i><font color="Green">E47</font></i></a>, <a class="ref" href="sppol_1.html#T63">SPPOL_1:63</a></i>;<br/>
<a NAME="E8:66"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="goboard7.html#E1:66"><i><font color="Green">E43</font></i></a>, <a class="ref" href="sppol_1.html#T36">SPPOL_1:36</a></i>;<br/>
<a NAME="E9:66"/>
( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">r</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  )
 
<b>by </b><i><a class="ref" href="goboard7.html#T5" target="_self">Th5</a>, <a class="txt" href="goboard7.html#E2:66"><i><font color="Green">E47</font></i></a>, <a class="ref" href="topreal1.html#T9">TOPREAL1:9</a></i>;<br/>
<a NAME="E10:66"/><b>then </b>
( <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">r</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  )
 
<b>by </b><i><a class="ref" href="goboard7.html#T5" target="_self">Th5</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E11:66"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="goboard7.html#T3" target="_self">Th3</a>, <a class="ref" href="goboard7.html#T6" target="_self">Th6</a>, <a class="ref" href="goboard7.html#T7" target="_self">Th7</a>, </i>;<br/>


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