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

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</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>let </b><font color="Maroon">B</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>set </b><font color="Maroon">f</font> =  <a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font>;<br/>
<b>assume </b><a NAME="E1:107"/><i><font color="Green">E47</font></i>: 
( <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="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; not <font color="Maroon">B</font> is <a href="jordan2c.html#V1" target="_self">Bounded</a> )
 ;<br/>

<a NAME="E2:107"/><i><font color="Green">E56</font></i>: 
 <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <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="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="goboard9.html#D1">GOBOARD9:def 1</a></i>;<br/>
<a NAME="E3:107"/><i><font color="Green">E57</font></i>: 
not  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> is <a href="jordan2c.html#V1" target="_self">Bounded</a>
 
<b>by </b><i><a class="ref" href="jordan2c.html#D3" target="_self">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon">B1</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="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E5:107"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B1</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="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="jordan2c.html#T18" target="_self">Th18</a>, <a class="ref" href="connsp_1.html#D6">CONNSP_1:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon">L1</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="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E7:107"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> &amp; <font color="Maroon">L1</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="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</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/>
<b>consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E9:107"/><i><font color="Green">E60</font></i>: 
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>  st <font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> holds <br/><span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">P</font> = <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> 2</span>)</span> <a href="subset_1.html#K6">\</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> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</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><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<a NAME="E11:107"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">P</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E2:107"><i><font color="Green">E56</font></i></a></i>;<br/>
<a NAME="E12:107"/>
<font color="Maroon">P</font> <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">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:107_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:107_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">q3</font> = <font color="Maroon">z</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> ;<br/><a NAME="E3:107_1_1"/><i><font color="Green">E65</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q3</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font>
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T24" target="_self">Th24</a></i>;<br/><a NAME="E4:107_1_1"/>
not <font color="Maroon">q3</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> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan2c.html#T23" target="_self">Th23</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E5:107_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:107_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</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">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T23" target="_self">Th23</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:107"/><b>then </b>
<font color="Maroon">P</font> <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">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E14:107"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">P</font>,<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="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 
;<br/>
<a NAME="E15:107"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="connsp_3.html#K4">Down</a> <font color="Maroon">P</font>,<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="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span> <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#T22" target="_self">Th22</a>, </i>;<br/>
<b>consider </b><font color="Maroon">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><b> such that </b><br/><a NAME="E17:107"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">q3</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q3</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r1</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E18:107"/>
<font color="Maroon">q3</font> <a href="hidden.html#R2">in</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>
 
;<br/>
<a NAME="E19:107"/><b>then </b><i><font color="Green">E131</font></i>: 
<font color="Maroon">q3</font> <a href="hidden.html#R2">in</a>  <a href="euclid.html#K1">REAL</a> 2
 
<b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:107_2"/>
<font color="Maroon">q3</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> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> </span>}</span> 
 ;<br/><a NAME="E2:107_2"/><b>then </b>
 ex <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> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q3</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> )
 ;<br/><b>hence </b><a NAME="E3:107_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E21:107"/><b>then </b>
<font color="Maroon">q3</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/>
<a NAME="E22:107"/><b>then </b><i><font color="Green">E152</font></i>: 
<font color="Maroon">L1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">P</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon">q4</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="E24:107"/><i><font color="Green">E154</font></i>: 
( <font color="Maroon">q4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q4</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r1</font> )
 <b>by </b><i><a class="ref" href="jordan2c.html#T18" target="_self">Th18</a>, </i>;<br/>
<a NAME="E25:107"/>
<font color="Maroon">q4</font> <a href="hidden.html#R2">in</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>
 
;<br/>
<a NAME="E26:107"/><b>then </b><i><font color="Green">E157</font></i>: 
<font color="Maroon">q4</font> <a href="hidden.html#R2">in</a>  <a href="euclid.html#K1">REAL</a> 2
 
<b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:107_3"/>
<font color="Maroon">q4</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> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> </span>}</span> 
 ;<br/><a NAME="E2:107_3"/><b>then </b>
 ex <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> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q4</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font> )
 ;<br/><b>hence </b><a NAME="E3:107_3"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan2c.html#T26" target="_self">Th26</a></i>;<br/></div><b>end;</b></div>
<a NAME="E28:107"/><b>then </b>
<font color="Maroon">q4</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/>
<a NAME="E29:107"/><b>then </b>
<font color="Maroon">B</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T26" target="_self">Th26</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E30:107"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T21" target="_self">Th21</a>, , <a class="ref" href="jordan2c.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan2c.html#T24" target="_self">Th24</a>, <a class="ref" href="jordan2c.html#T25" target="_self">Th25</a>, <a class="ref" href="jordan2c.html#T17" target="_self">Th17</a></i>;<br/>


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