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<b>let </b><font color="Maroon">s1</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">t1</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">s2</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">t2</font> be   <a href="real_1.html#NM1">Real</a>;<br/><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>assume </b><a NAME="E1:35"/>
<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">s</font>,<font color="Olive">t</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t</font> is   <a href="real_1.html#NM1">Real</a> : (  not <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> or  not <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> or  not <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> or  not <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:35"/><b>then </b><i><font color="Green">E28</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"><span class="p3">{<span class="default"> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s3</font>,<font color="Olive">t3</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s3</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">s3</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p3">{<span class="default"> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s4</font>,<font color="Olive">t4</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s4</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t4</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">t4</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t1</font> </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s5</font>,<font color="Olive">t5</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s5</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t5</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">s2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s5</font> </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s6</font>,<font color="Olive">t6</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s6</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t6</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">t2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">t6</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">A0</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s</font>,<font color="Olive">t</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</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>, </i>;<br/>
<b>reconsider </b><font color="Maroon">A1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s</font>,<font color="Olive">t</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t1</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>, </i>;<br/>
<b>reconsider </b><font color="Maroon">A2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s</font>,<font color="Olive">t</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">s2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</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>, </i>;<br/>
<b>reconsider </b><font color="Maroon">A3</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s</font>,<font color="Olive">t</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">t2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">t</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>, </i>;<br/>
<a NAME="E7:35"/>
( <font color="Maroon">A0</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A1</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i><a class="txt" href="jordan1.html#E10:35"><i><font color="Green">E31</font></i></a>, </i>;<br/>
<a NAME="E8:35"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">A0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A1</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T37">TOPS_1:37</a></i>;<br/>
<a NAME="E9:35"/>
<font color="Maroon">A2</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="jordan1.html#E8:35"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E10:35"/><b>then </b><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">A0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="tops_1.html#T37">TOPS_1:37</a></i>;<br/>
<a NAME="E11:35"/>
<font color="Maroon">A3</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E12:35"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, , <a class="ref" href="tops_1.html#T37">TOPS_1:37</a></i>;<br/>


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