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<b>let </b><font color="Maroon" title="c1">p1</font>, <font color="Maroon" title="c2">p2</font>, <font color="Maroon" title="c3">q1</font>, <font color="Maroon" title="c4">q2</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><a NAME="E1:50"/><i><font color="Green" title="E43">A1</font></i>: 
(  <a href="jordan3.html#R2" title="JORDAN3:pred.2">LE</a> <font color="Maroon" title="c3">q1</font>,<font color="Maroon" title="c4">q2</font>,<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font> &amp;  <a href="jordan3.html#R2" title="JORDAN3:pred.2">LE</a> <font color="Maroon" title="c4">q2</font>,<font color="Maroon" title="c3">q1</font>,<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font> )
 ;<br/>

<a NAME="E2:50"/><b>then </b><i><font color="Green" title="E44">A2</font></i>: 
( <font color="Maroon" title="c3">q1</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">p1</font>,<font color="Maroon" title="c2">p2</font> &amp; <font color="Maroon" title="c4">q2</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">p1</font>,<font color="Maroon" title="c2">p2</font> &amp; ( for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r1</font> &amp; <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c3">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Olive" title="b1">r1</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r1</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p2</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">r2</font> &amp; <font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c4">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Olive" title="b2">r2</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">r2</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p2</font></span>)</span> holds <br/><font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">r2</font> ) )
 
<b>by </b><i><a class="ref" href="jordan3.html#D6" target="_self" title="JORDAN3:def.6">Def6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:50"/><i><font color="Green" title="E45">A3</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r1</font> &amp; <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c3">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r1</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r1</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="sppol_1.html#T21" title="SPPOL_1:th.21">SPPOL_1:21</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">r2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E6:50"/><i><font color="Green" title="E46">A4</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r2</font> &amp; <font color="Maroon" title="c6">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c4">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c6">r2</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">r2</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan3.html#E2:50"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="sppol_1.html#T21" title="SPPOL_1:th.21">SPPOL_1:21</a></i>;<br/>
<a NAME="E7:50"/><i><font color="Green" title="E47">A5</font></i>: 
<font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r2</font>
 
<b>by </b><i><a class="txt" href="jordan3.html#E1:50"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="jordan3.html#E4:50"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="jordan3.html#E6:50"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="jordan3.html#D6" target="_self" title="JORDAN3:def.6">Def6</a></i>;<br/>
<a NAME="E8:50"/>
<font color="Maroon" title="c6">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r1</font>
 
<b>by </b><i><a class="txt" href="jordan3.html#E1:50"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="jordan3.html#E4:50"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="jordan3.html#E6:50"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="jordan3.html#D6" target="_self" title="JORDAN3:def.6">Def6</a></i>;<br/>
<a NAME="E9:50"/><b>then </b>
<font color="Maroon" title="c5">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r2</font>
 
<b>by </b><i><a class="txt" href="jordan3.html#E7:50"><i><font color="Green" title="E47">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E10:50"/>
<font color="Maroon" title="c3">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q2</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E4:50"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="jordan3.html#E6:50"><i><font color="Green" title="E46">A4</font></i></a></i>;<br/>


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