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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></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="E1:66"/><i><font color="Green">E57</font></i>: 
(  <a href="jordan3.html#R2" target="_self">LE</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="jordan3.html#R2" target="_self">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E2:66"/><b>then </b><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">c<sub>3</sub></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">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">c<sub>4</sub></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">b<sub>2</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> ) )
 
<b>by </b><i><a class="ref" href="jordan3.html#D6" target="_self">Def6</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:66"/><i><font color="Green">E59</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">c<sub>3</sub></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">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:66"/><i><font color="Green">E60</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">c<sub>4</sub></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">c<sub>6</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan3.html#E2:66"><i><font color="Green">E58</font></i></a>, <a class="ref" href="sppol_1.html#T21">SPPOL_1:21</a></i>;<br/>
<a NAME="E7:66"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="jordan3.html#E1:66"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan3.html#E4:66"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan3.html#E6:66"><i><font color="Green">E60</font></i></a>, <a class="ref" href="jordan3.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E8:66"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="jordan3.html#E1:66"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan3.html#E4:66"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan3.html#E6:66"><i><font color="Green">E60</font></i></a>, <a class="ref" href="jordan3.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E9:66"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="jordan3.html#E7:66"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E10:66"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jordan3.html#E4:66"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan3.html#E6:66"><i><font color="Green">E60</font></i></a></i>;<br/>


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