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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">C</font> be  <a href="connsp_1.html#V2">connected</a> <a href="compts_1.html#V6">compact</a> non <a href="sppol_1.html#V1">horizontal</a> non <a href="sppol_1.html#V2">vertical</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>;<br/>



<b>set </b><font color="Maroon">f</font> =  <a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>;<br/>
<a NAME="E1:15"/><i><font color="Green">E40</font></i>: 
 <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> 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> : <font color="Olive">B</font> <a href="jordan2c.html#R2">is_outside_component_of</a> <font color="Maroon">C</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="jordan2c.html#D6">JORDAN2C:def 6</a></i>;<br/>
<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="E2:15"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E3:15"/>
 <a href="jordan2c.html#K2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> 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> : <font color="Olive">B</font> <a href="jordan2c.html#R2">is_outside_component_of</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> </span>}</span> 
 
<b>by </b><i><a class="ref" href="jordan2c.html#D6">JORDAN2C:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">L</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:15"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E6:15"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> 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> : <font color="Olive">B</font> <a href="jordan2c.html#R2">is_outside_component_of</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">B</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><b> such that </b><br/><a NAME="E8:15"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E9:15"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">B</font> <a href="jordan2c.html#R2">is_outside_component_of</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">B1</font> = <font color="Maroon">B</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<a NAME="E11:15"/>
<font color="Maroon">B</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i>, <a class="ref" href="jordan2c.html#D4">JORDAN2C:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">B2</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"><span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E13:15"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">B2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E14:15"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">B2</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"><span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="connsp_1.html#D6">CONNSP_1:def 6</a></i>;<br/>
<a NAME="E15:15"/>
<font color="Maroon">B2</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i>, <a class="ref" href="connsp_1.html#D5">CONNSP_1:def 5</a></i>;<br/>
<a NAME="E16:15"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">B</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i>, <a class="ref" href="connsp_1.html#T24">CONNSP_1:24</a></i>;<br/>
<a NAME="E17:15"/>
not <font color="Maroon">B</font> is <a href="jordan2c.html#V1">Bounded</a>
 
<b>by </b><i>, <a class="ref" href="jordan2c.html#D4">JORDAN2C:def 4</a></i>;<br/>
<a NAME="E18:15"/><b>then </b><i><font color="Green">E61</font></i>: 
not <font color="Maroon">B1</font> is <a href="tbsp_1.html#V6">bounded</a>
 
<b>by </b><i><a class="ref" href="jordan2c.html#D2">JORDAN2C:def 2</a></i>;<br/>
<a NAME="E19:15"/>
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> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> 
 
<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">P1</font> =  <a href="connsp_3.html#K1">Component_of</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K4">Down</a> <font color="Maroon">B</font>,<span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></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><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E21:15"/>
 <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="jordan2c.html#R1">is_inside_component_of</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="gobrd14.html#T45">GOBRD14:45</a></i>;<br/>
<a NAME="E22:15"/><b>then </b><i><font color="Green">E62</font></i>: 
( <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">B1</font> <a href="xboole_0.html#R1">misses</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="jordan2c.html#T126">JORDAN2C:126</a></i>;<br/>
<a NAME="E23:15"/><b>then </b>
<font color="Maroon">B1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E24:15"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">P1</font> <a href="jordan2c.html#R2">is_outside_component_of</a> <font color="Maroon">C</font>
 
<b>by </b><i>, , <a class="ref" href="jordan2c.html#T71">JORDAN2C:71</a></i>;<br/>
<a NAME="E25:15"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">B1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3">RightComp</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E26:15"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:15_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:15_1"/><b>then </b>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</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="E3:15_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E27:15"/><b>then </b>
 <a href="connsp_3.html#K4">Down</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E28:15"/><b>then </b>
 <a href="connsp_3.html#K4">Down</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i>, <a class="ref" href="jordan2c.html#T15">JORDAN2C:15</a></i>;<br/>
<a NAME="E29:15"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="connsp_3.html#K4">Down</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="connsp_3.html#K1">Component_of</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K4">Down</a> <font color="Maroon">B</font>,<span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="connsp_3.html#T1">CONNSP_3:1</a></i>;<br/>
<a NAME="E30:15"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i>, , , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E31:15"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E32:15"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E33:15"/>
<font color="Maroon">P1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">W</font> where <font color="Olive">W</font> 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> : <font color="Olive">W</font> <a href="jordan2c.html#R2">is_outside_component_of</a> <font color="Maroon">C</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E34:15"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div>
