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<b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</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">r3</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">q3</font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><a NAME="E1:109"/><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"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r1</font>,<font color="Olive">r2</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r1</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r2</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r1</font> &amp; <font color="Olive">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q3</font> ) </span>}</span> 
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="real_1.html#NM1">Real</a>,  <a href="real_1.html#NM1">Real</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font></span><a href="euclid.html#K23">]|</a></span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="real_1.html#NM1">Real</a>,  <a href="real_1.html#NM1">Real</a>] <b>means </b><font color="Maroon">r3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="real_1.html#NM1">Real</a>,  <a href="real_1.html#NM1">Real</a>] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q3</font>;<br/>
<a NAME="E2:109"/><i><font color="Green">E31</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">r1</font>,<font color="Olive">r2</font>) where <font color="Olive">r1</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r2</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r1</font>,<font color="Olive">r2</font>] </span>}</span>  is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>from </b><i><a class="ref" href="domain_1.html#S9">DOMAIN_1:sch 9</a>();<br/></i>
<a NAME="E3:109"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">r1</font>,<font color="Olive">r2</font>) where <font color="Olive">r1</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r2</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">r1</font>,<font color="Olive">r2</font>] </span>}</span>  is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>from </b><i><a class="ref" href="domain_1.html#S9">DOMAIN_1:sch 9</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">Q1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r1</font>,<font color="Olive">r2</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r1</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r2</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">r3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r1</font> </span>}</span> , <font color="Maroon">Q2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r1</font>,<font color="Olive">r2</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r1</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r2</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q3</font> </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/>
<a NAME="E5:109"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">Q1</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="jordan1.html#T25">JORDAN1:25</a></i>;<br/>
<a NAME="E6:109"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">Q2</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="jordan1.html#T26">JORDAN1:26</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:109_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/><b>then consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:109_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">r3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q3</font> )
 <b>by </b><i/>;<br/><a NAME="E4:109_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:109_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:109_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q2</font>
 ;<br/><a NAME="E3:109_1"/><b>then </b><i><font color="Green">E36</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:109_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">r3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> )
 ;<br/><b>consider </b><font color="Maroon">r1'</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">r2'</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:109_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r1'</font>,<font color="Maroon">r2'</font></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">r1'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q3</font> )
 <b>by </b><i/>;<br/><a NAME="E8:109_1"/>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r1'</font>,<font color="Maroon">r2'</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">r1'</font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r1'</font>,<font color="Maroon">r2'</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">r2'</font> )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E9:109_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="pscomp_1.html#T4" target="_self">Th4</a>, </i>;<br/></div><b>end;</b></div>
<a NAME="E8:109"/><b>then </b>
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<b>hence </b><a NAME="E9:109"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="pscomp_1.html#T3" target="_self">Th3</a>, , <a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/>


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