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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">s2</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">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><b>that </b><br/><a NAME="E1:57"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t2</font> )
 <b>and </b><br/><a NAME="E2:57"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t1</font> ) or ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s1</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> ) or ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s1</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font> ) or ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t1</font> ) ) </span>}</span> 
 ;<br/>

<b>reconsider </b><font color="Maroon">P1</font> = <span class="p1">{<span class="default"> <font color="Olive">pa</font> where <font color="Olive">pa</font> is   <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> : ( <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">pa</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">pa</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">pa</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">pa</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t2</font> ) </span>}</span> , <font color="Maroon">P2</font> = <span class="p1">{<span class="default"> <font color="Olive">pc</font> where <font color="Olive">pc</font> is   <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> : (  not <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">pc</font> <a href="euclid.html#K21">`1</a>  or  not <font color="Olive">pc</font> <a href="euclid.html#K21">`1</a>  <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">pc</font> <a href="euclid.html#K22">`2</a>  or  not <font color="Olive">pc</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</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">PC</font> = <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  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> ;<br/>
<a NAME="E5:57"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">pa</font> where <font color="Olive">pa</font> is   <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> : ( <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">pa</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">pa</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">pa</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">pa</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t2</font> ) </span>}</span>  &amp; <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">pc</font> where <font color="Olive">pc</font> is   <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> : (  not <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">pc</font> <a href="euclid.html#K21">`1</a>  or  not <font color="Olive">pc</font> <a href="euclid.html#K21">`1</a>  <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">pc</font> <a href="euclid.html#K22">`2</a>  or  not <font color="Olive">pc</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> ) </span>}</span>  )
 
;<br/>
<a NAME="E6:57"/><b>then </b><i><font color="Green">E32</font></i>: 
( <font color="Maroon">PC</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P2</font> &amp; <font color="Maroon">PC</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">P1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">P2</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E7:57"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P1</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">P1</font> &amp; <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P2</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">P2</font> )
 
<b>by </b><i>, , , <a class="ref" href="jordan1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E8:57"/>
for <font color="Olive">P1A</font>, <font color="Olive">P2B</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>  st <font color="Olive">P1A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> &amp; <font color="Olive">P2B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font> holds <br/>( <font color="Olive">P1A</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Olive">P2B</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E9:57"/>
<font color="Maroon">P</font> <a href="jordan1.html#NR1" target="_self">has_property_J</a> 
 <b>by </b><i>, , </i>;<br/>


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