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

<b>let </b><font color="Maroon">p</font> be   <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>;<br/><b>let </b><font color="Maroon">R</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">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="E1:32"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">R</font> <a href="topreal4.html#NR3" target="_self">is_Region</a> 
 <b>and </b><br/><a NAME="E2:32"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E3:32"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</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">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> or  ex <font color="Olive">P1</font> being  <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> st <br/>( <font color="Olive">P1</font> <a href="topreal4.html#R1" target="_self">is_S-P_arc_joining</a> <font color="Maroon">p</font>,<font color="Olive">q</font> &amp; <font color="Olive">P1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> ) ) </span>}</span> 
 ;<br/>

<b>reconsider </b><font color="Maroon">R'</font> = <font color="Maroon">R</font> as  non <a href="xboole_0.html#V1">empty</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/>;<br/>
<a NAME="E5:32"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font>
 
<b>by </b><i>, , </i>;<br/>
<b>set </b><font color="Maroon">P2</font> = <font color="Maroon">R</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">P</font>;<br/>
<a NAME="E6:32"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">R</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><i><font color="Green">E60</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"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:32_1_1"/><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">R</font> &amp; not <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="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">q2</font> = <font color="Maroon">x</font> as   <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>by </b><i/>;<br/><a NAME="E4:32_1_1"/>
( <font color="Maroon">q2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; ( for <font color="Olive">P1</font> being  <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>  holds <br/>(  not <font color="Olive">P1</font> <a href="topreal4.html#R1" target="_self">is_S-P_arc_joining</a> <font color="Maroon">p</font>,<font color="Maroon">q2</font> or  not <font color="Olive">P1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> ) ) )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:32_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</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">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; ( for <font color="Olive">P1</font> being  <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>  holds <br/>(  not <font color="Olive">P1</font> <a href="topreal4.html#R1" target="_self">is_S-P_arc_joining</a> <font color="Maroon">p</font>,<font color="Olive">q</font> or  not <font color="Olive">P1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</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">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</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">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; ( for <font color="Olive">P1</font> being  <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>  holds <br/>(  not <font color="Olive">P1</font> <a href="topreal4.html#R1" target="_self">is_S-P_arc_joining</a> <font color="Maroon">p</font>,<font color="Olive">q</font> or  not <font color="Olive">P1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> ) ) ) </span>}</span> 
 ;<br/><a NAME="E2:32_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
 ex <font color="Olive">q3</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> st <br/>( <font color="Olive">q3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">q3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> &amp; <font color="Olive">q3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; ( for <font color="Olive">P1</font> being  <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>  holds <br/>(  not <font color="Olive">P1</font> <a href="topreal4.html#R1" target="_self">is_S-P_arc_joining</a> <font color="Maroon">p</font>,<font color="Olive">q3</font> or  not <font color="Olive">P1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> ) ) )
 ;<br/><a NAME="E3:32_1_2"/><b>then </b>
for <font color="Olive">q</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>  holds <br/>(  not <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or ( not <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; ( for <font color="Olive">P1</font> being  <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>  holds <br/>(  not <font color="Olive">P1</font> <a href="topreal4.html#R1" target="_self">is_S-P_arc_joining</a> <font color="Maroon">p</font>,<font color="Olive">q</font> or  not <font color="Olive">P1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> ) ) ) )
 ;<br/><a NAME="E4:32_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:32_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:32_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">P</font> iff <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</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">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; ( for <font color="Olive">P1</font> being  <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>  holds <br/>(  not <font color="Olive">P1</font> <a href="topreal4.html#R1" target="_self">is_S-P_arc_joining</a> <font color="Maroon">p</font>,<font color="Olive">q</font> or  not <font color="Olive">P1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> ) ) ) </span>}</span>  )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E8:32"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">R</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</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">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; ( for <font color="Olive">P1</font> being  <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>  holds <br/>(  not <font color="Olive">P1</font> <a href="topreal4.html#R1" target="_self">is_S-P_arc_joining</a> <font color="Maroon">p</font>,<font color="Olive">q</font> or  not <font color="Olive">P1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> ) ) ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P22</font> = <font color="Maroon">R</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">P</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> ;<br/>
<a NAME="E10:32"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">P22</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E11:32"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E12:32"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i/>;<br/>
<a NAME="E13:32"/>
<font color="Maroon">R</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:32"/><b>then </b><i><font color="Green">E72</font></i>: 
<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">R'</font> is <a href="connsp_1.html#V1">connected</a>
 
<b>by </b><i><a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a></i>;<br/>
<a NAME="E15:32"/><i><font color="Green">E73</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <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">R</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">P11</font> = <font color="Maroon">P</font>, <font color="Maroon">P12</font> = <font color="Maroon">P22</font> as   <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">R</font></span>)</span> <b>by </b><i>, , <a class="ref" href="pre_topc.html#T16">PRE_TOPC:16</a></i>;<br/>
<a NAME="E17:32"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">P11</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">P12</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E18:32"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">P11</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P12</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <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">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E19:32"/>
<font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>;<br/>
<a NAME="E20:32"/><b>then </b><i><font color="Green">E76</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <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">R</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P11</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P12</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>
<a NAME="E21:32"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">P22</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> &amp; <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</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="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E22:32"/>
( <font color="Maroon">P11</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</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">R</font></span>)</span></span>)</span> &amp; <font color="Maroon">P12</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P22</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">R</font></span>)</span></span>)</span> )
 
<b>by </b><i>, , , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E23:32"/><b>then </b>
( <font color="Maroon">P11</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</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">R</font></span>)</span> &amp; <font color="Maroon">P12</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</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">R</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<a NAME="E24:32"/><b>then </b>
( <font color="Maroon">P11</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">P12</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E25:32"/><b>then </b>
<font color="Maroon">R</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , , , , <a class="ref" href="connsp_1.html#T12">CONNSP_1:12</a></i>;<br/>
<b>hence </b><a NAME="E26:32"/>
<font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>


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