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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">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>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:3"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">p</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">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:3"/><i><font color="Green">E28</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>and </b><br/><a NAME="E4:3"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">p</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">r</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">r</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/>
<i><font color="Green">E32</font></i>: <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="topreal3.html#T7">TOPREAL3:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T9">TOPREAL3:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T9">TOPREAL3:9</a></i>
;<br/>
<a NAME="E6:3"/>
1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E7:3"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E8:3"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E9:3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
;<br/>
<b>hence </b><a NAME="E10:3"/>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font>
 <b>by </b><i>, , , <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>


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