<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">B0</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">K0</font> be   <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> <font color="Maroon">B0</font></span>)</span>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B0</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K0</font></span>)</span>,<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> <font color="Maroon">B0</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:27"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <a href="jgraph_3.html#K1" target="_self">Sq_Circ</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">K0</font> &amp; <font color="Maroon">B0</font> <a href="hidden.html#R1">=</a> 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> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">K0</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="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) ) &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span>  )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <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>] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> ) );<br/>
<b>set </b><font color="Maroon">k0</font> = <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="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>] &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">b0</font> = 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> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span>;<br/>
<a NAME="E2:27"/>
the <a href="struct_0.html#U1">carrier</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> <font color="Maroon">B0</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B0</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">K1</font> = <font color="Maroon">K0</font> 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><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <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>] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> ) );<br/>
<a NAME="E4:27"/>
<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="Maroon">S<sub>2</sub></font>[<font color="Olive">p</font>] &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span>  <a href="tarski.html#R1">c=</a> 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> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>from </b><i><a class="ref" href="jgraph_3.html#S1" target="_self">JGRAPH_3:sch 1</a>();<br/></i>
<a NAME="E5:27"/><b>then </b><i><font color="Green">E39</font></i>: 
<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> <font color="Maroon">B0</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">K0</font> <a href="hidden.html#R1">=</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> <font color="Maroon">K1</font>
 
<b>by </b><i><a class="txt" href="jgraph_3.html#E1:27"><i><font color="Green">E38</font></i></a>, <a class="ref" href="pre_topc.html#T28">PRE_TOPC:28</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <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>] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span> ) );<br/>
<b>reconsider </b><font color="Maroon">K1</font> = <span class="p1">{<span class="default"> <font color="Olive">p7</font> where <font color="Olive">p7</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">S<sub>3</sub></font>[<font color="Olive">p7</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>from </b><i><a class="ref" href="jgraph_2.html#S1">JGRAPH_2:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">K2</font> = <span class="p1">{<span class="default"> <font color="Olive">p7</font> where <font color="Olive">p7</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">p7</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p7</font> <a href="euclid.html#K22">`2</a>  </span>}</span>  as  <a href="pre_topc.html#V4">closed</a> <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><a class="ref" href="jgraph_2.html#T56">JGRAPH_2:56</a></i>;<br/>
<b>reconsider </b><font color="Maroon">K3</font> = <span class="p1">{<span class="default"> <font color="Olive">p7</font> where <font color="Olive">p7</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> :  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p7</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p7</font> <a href="euclid.html#K21">`1</a>  </span>}</span>  as  <a href="pre_topc.html#V4">closed</a> <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><a class="ref" href="jgraph_2.html#T58">JGRAPH_2:58</a></i>;<br/>
<b>reconsider </b><font color="Maroon">K4</font> = <span class="p1">{<span class="default"> <font color="Olive">p7</font> where <font color="Olive">p7</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">p7</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p7</font> <a href="euclid.html#K21">`1</a>  </span>}</span>  as  <a href="pre_topc.html#V4">closed</a> <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><a class="ref" href="jgraph_2.html#T56">JGRAPH_2:56</a></i>;<br/>
<b>reconsider </b><font color="Maroon">K5</font> = <span class="p1">{<span class="default"> <font color="Olive">p7</font> where <font color="Olive">p7</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">p7</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Olive">p7</font> <a href="euclid.html#K22">`2</a> </span>)</span> </span>}</span>  as  <a href="pre_topc.html#V4">closed</a> <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><a class="ref" href="jgraph_2.html#T58">JGRAPH_2:58</a></i>;<br/>
<a NAME="E11:27"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">K2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K3</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T35">TOPS_1:35</a></i>;<br/>
<a NAME="E12:27"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">K4</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K5</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T35">TOPS_1:35</a></i>;<br/>
<a NAME="E13:27"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">K2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K3</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">K4</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K5</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">K1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:27_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">K2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K3</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">K4</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K5</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K3</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K4</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K5</font> )
 </span><b>by </b><i><a class="txt" href="jgraph_3.html#E1:27_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K3</font>
 ;<br/><div class="add"><a NAME="E2:27_1_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K3</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">p7</font> being   <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><b> such that </b><br/><a NAME="E4:27_1_1_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">p7</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p7</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p7</font> <a href="euclid.html#K22">`2</a>  )
 ;<br/><b>consider </b><font color="Maroon">p8</font> being   <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><b> such that </b><br/><a NAME="E6:27_1_1_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">p8</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="ref" href="jgraph_3.html#T12" target="_self">Th12</a></i>;<br/><b>thus </b><a NAME="E7:27_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K1</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K4</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K5</font>
 ;<br/><div class="add"><a NAME="E2:27_1_1_2"/><b>then </b><i><font color="Green">E62</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K4</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K5</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">p7</font> being   <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><b> such that </b><br/><a NAME="E4:27_1_1_2"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">p7</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p7</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p7</font> <a href="euclid.html#K22">`2</a>  )
 ;<br/><b>consider </b><font color="Maroon">p8</font> being   <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><b> such that </b><br/><a NAME="E6:27_1_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">p8</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p8</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="jgraph_3.html#T12" target="_self">Th12</a></i>;<br/><b>thus </b><a NAME="E7:27_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K1</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E14:27"/>
<font color="Maroon">K1</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">K2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K3</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">K4</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K5</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:27_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <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><b> such that </b><br/><a NAME="E3:27_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; ( ( <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  ) or ( <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> ) ) )
 ;<br/>
<a NAME="E4:27_2"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K3</font> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K4</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K5</font> ) )
 
<b>by </b><i/>;<br/>
<a NAME="E5:27_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K3</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K4</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K5</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:27_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">K2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K3</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">K4</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K5</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:27"/><b>then </b>
<font color="Maroon">K1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">K2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K3</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">K4</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K5</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_3.html#E13:27"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E16:27"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">K1</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="jgraph_3.html#E11:27"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jgraph_3.html#E12:27"><i><font color="Green">E41</font></i></a>, <a class="ref" href="tops_1.html#T36">TOPS_1:36</a></i>;<br/>
<a NAME="E17:27"/>
<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="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>] &amp; <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2 ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p7</font> where <font color="Olive">p7</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">S<sub>1</sub></font>[<font color="Olive">p7</font>] </span>}</span>  <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>from </b><i><a class="ref" href="jgraph_3.html#S2" target="_self">JGRAPH_3:sch 2</a>();<br/></i>
<a NAME="E18:27"/><b>then </b>
<font color="Maroon">K0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B0</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_3.html#E1:27"><i><font color="Green">E38</font></i></a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>hence </b><a NAME="E19:27"/>
( <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">K0</font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i><a class="txt" href="jgraph_3.html#E1:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jgraph_3.html#E5:27"><i><font color="Green">E39</font></i></a>, , , <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/>


</div>
