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<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>, <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>, <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>, <font color="Maroon">q2</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="E1:52"/><i><font color="Green">E50</font></i>: 
(  <a href="jordan3.html#R2" target="_self">LE</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan3.html#R2" target="_self">LE</a> <font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/>

<a NAME="E2:52"/><b>then </b><i><font color="Green">E51</font></i>: 
( <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; <font color="Maroon">q2</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; ( for <font color="Olive">r1</font>, <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r1</font> &amp; <font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r2</font> &amp; <font color="Olive">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> holds <br/><font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r2</font> ) )
 
<b>by </b><i><a class="txt" href="jordan3.html#E6:52"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:52"/><i><font color="Green">E52</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="sppol_1.html#T21">SPPOL_1:21</a></i>;<br/>
<b>consider </b><font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:52"/><i><font color="Green">E53</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="sppol_1.html#T21">SPPOL_1:21</a></i>;<br/>
<a NAME="E7:52"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font>
 
<b>by </b><i><a class="ref" href="jordan3.html#T17" target="_self">Th17</a>, <a class="ref" href="jordan3.html#T18" target="_self">Th18</a>, <a class="ref" href="jordan3.html#T19" target="_self">Th19</a>, <a class="txt" href="jordan3.html#E6:52"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E8:52"/>
<font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font>
 
<b>by </b><i><a class="ref" href="jordan3.html#T17" target="_self">Th17</a>, <a class="ref" href="jordan3.html#T18" target="_self">Th18</a>, <a class="ref" href="jordan3.html#T19" target="_self">Th19</a>, <a class="txt" href="jordan3.html#E6:52"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E9:52"/><b>then </b>
<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font>
 
<b>by </b><i><a class="ref" href="jordan3.html#T21" target="_self">Th21</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E10:52"/>
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font>
 <b>by </b><i><a class="ref" href="jordan3.html#T18" target="_self">Th18</a>, <a class="ref" href="jordan3.html#T19" target="_self">Th19</a></i>;<br/>


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