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<b>let </b><font color="Maroon" title="c1">s1</font>, <font color="Maroon" title="c2">s2</font>, <font color="Maroon" title="c3">t1</font>, <font color="Maroon" title="c4">t2</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>







<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><i><font color="Green" title="E31">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">qc</font> where <font color="Olive" title="b1">qc</font> is   <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> : (  not <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">qc</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or  not <font color="Olive" title="b1">qc</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s2</font> or  not <font color="Maroon" title="c3">t1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">qc</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  or  not <font color="Olive" title="b1">qc</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t2</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c6">q</font> being   <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><b> such that </b><br/><a NAME="E3:32_1_1"/><i><font color="Green" title="E32">A2</font></i>: 
( <font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> &amp; (  not <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or  not <font color="Maroon" title="c6">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s2</font> or  not <font color="Maroon" title="c3">t1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  or  not <font color="Maroon" title="c6">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t2</font> ) )
 ;<br/><a NAME="E4:32_1_1"/>
( <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> &amp; (  not <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or  not <font color="Maroon" title="c6">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s2</font> or  not <font color="Maroon" title="c3">t1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  or  not <font color="Maroon" title="c6">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t2</font> ) )
 <b>by </b><i><a class="txt" href="jordan1.html#E3:32_1_1"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><b>hence </b><a NAME="E5:32_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">s2a</font>,<font color="Olive" title="b2">t2a</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">s2a</font>, <font color="Olive" title="b2">t2a</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  not <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">s2a</font> or  not <font color="Olive" title="b1">s2a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s2</font> or  not <font color="Maroon" title="c3">t1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">t2a</font> or  not <font color="Olive" title="b2">t2a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t2</font> ) </span>}</span> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">sb</font>,<font color="Olive" title="b2">tb</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">sb</font>, <font color="Olive" title="b2">tb</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  not <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">sb</font> or  not <font color="Olive" title="b1">sb</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s2</font> or  not <font color="Maroon" title="c3">t1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">tb</font> or  not <font color="Olive" title="b2">tb</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t2</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c6">sb</font>, <font color="Maroon" title="c7">tb</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:32_1_2"/><i><font color="Green" title="E32">A3</font></i>: 
( <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c6">sb</font>,<font color="Maroon" title="c7">tb</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> &amp; (  not <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">sb</font> or  not <font color="Maroon" title="c6">sb</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s2</font> or  not <font color="Maroon" title="c3">t1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">tb</font> or  not <font color="Maroon" title="c7">tb</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t2</font> ) )
 ;<br/><b>set </b><font color="Maroon" title="c8">qa</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c6">sb</font>,<font color="Maroon" title="c7">tb</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>;<br/><a NAME="E4:32_1_2"/>
( <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c6">sb</font>,<font color="Maroon" title="c7">tb</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> &amp; (  not <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c6">sb</font>,<font color="Maroon" title="c7">tb</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or  not <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c6">sb</font>,<font color="Maroon" title="c7">tb</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s2</font> or  not <font color="Maroon" title="c3">t1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c6">sb</font>,<font color="Maroon" title="c7">tb</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  or  not <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c6">sb</font>,<font color="Maroon" title="c7">tb</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t2</font> ) )
 <b>by </b><i><a class="txt" href="jordan1.html#E3:32_1_2"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E5:32_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">qc</font> where <font color="Olive" title="b1">qc</font> is   <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> : (  not <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">qc</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or  not <font color="Olive" title="b1">qc</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s2</font> or  not <font color="Maroon" title="c3">t1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">qc</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  or  not <font color="Olive" title="b1">qc</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t2</font> ) </span>}</span> 
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:32_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">qc</font> where <font color="Olive" title="b1">qc</font> is   <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> : (  not <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">qc</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or  not <font color="Olive" title="b1">qc</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s2</font> or  not <font color="Maroon" title="c3">t1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">qc</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  or  not <font color="Olive" title="b1">qc</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t2</font> ) </span>}</span>  iff <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">sb</font>,<font color="Olive" title="b2">tb</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">sb</font>, <font color="Olive" title="b2">tb</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  not <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">sb</font> or  not <font color="Olive" title="b1">sb</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s2</font> or  not <font color="Maroon" title="c3">t1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">tb</font> or  not <font color="Olive" title="b2">tb</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t2</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="jordan1.html#E1:32_1"><i><font color="Green" title="E31">A1</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:32"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">qc</font> where <font color="Olive" title="b1">qc</font> is   <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> : (  not <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">qc</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or  not <font color="Olive" title="b1">qc</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s2</font> or  not <font color="Maroon" title="c3">t1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">qc</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  or  not <font color="Olive" title="b1">qc</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t2</font> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">sb</font>,<font color="Olive" title="b2">tb</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">sb</font>, <font color="Olive" title="b2">tb</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  not <font color="Maroon" title="c1">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">sb</font> or  not <font color="Olive" title="b1">sb</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s2</font> or  not <font color="Maroon" title="c3">t1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">tb</font> or  not <font color="Olive" title="b2">tb</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t2</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


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