<?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">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><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>assume </b><a NAME="E1:125"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">p</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>set </b><font color="Maroon">A</font> =  <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/>
<a NAME="E2:125"/>
<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>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="sprect_1.html#D3">SPRECT_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">A1</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>, <font color="Maroon">A2</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="E4:125"/><i><font color="Green">E56</font></i>: 
( <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>  <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">A1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A2</font> &amp; <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font> &amp;  <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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font> &amp; ( for <font color="Olive">C1</font>, <font color="Olive">C2</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>  st <font color="Olive">C1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> &amp; <font color="Olive">C2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> holds <br/>( <font color="Olive">C1</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> &amp; <font color="Olive">C2</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/>;<br/>
<a NAME="E5:125"/><i><font color="Green">E57</font></i>: 
 <a href="connsp_3.html#K4">Down</a> <font color="Maroon">A1</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> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T28" target="_self">Th28</a>, <a class="ref" href="xboole_1.html#T21">XBOOLE_1:21</a></i>;<br/>
<a NAME="E6:125"/><i><font color="Green">E58</font></i>: 
 <a href="connsp_3.html#K4">Down</a> <font color="Maroon">A2</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> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T28" target="_self">Th28</a>, <a class="ref" href="xboole_1.html#T21">XBOOLE_1:21</a></i>;<br/>
<a NAME="E7:125"/><b>then </b><i><font color="Green">E59</font></i>: 
(  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">A1</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> <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> &amp;  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">A2</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> <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#T28" target="_self">Th28</a>, <a class="ref" href="jordan2c.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E8:125"/>
 <a href="jordan2c.html#K1" target="_self">BDD</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="jordan2c.html#R1" target="_self">is_inside_component_of</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>
 
<b>by </b><i/>;<br/>
<a NAME="E9:125"/><b>then </b>
 <a href="jordan2c.html#K1" target="_self">BDD</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="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/>;<br/>
<b>then 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="E11:125"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="jordan2c.html#K1" target="_self">BDD</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> &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="connsp_1.html#D6">CONNSP_1:def 6</a></i>;<br/>
<a NAME="E12:125"/>
<font color="Maroon">B1</font> <a href="tarski.html#R1">c=</a> 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"><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>
 
;<br/>
<a NAME="E13:125"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="jordan2c.html#K1" target="_self">BDD</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="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T28" target="_self">Th28</a>, , <a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>consider </b><font color="Maroon">q1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="jordan2c.html#K1" target="_self">BDD</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>;<br/>
<a NAME="E15:125"/><i><font color="Green">E63</font></i>: 
 <a href="jordan2c.html#K1" target="_self">BDD</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="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E16:125"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">q1</font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K1" target="_self">BDD</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>
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:125_1"/>
( <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> or <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> )
 </span><b>by </b><i><a class="ref" href="jordan2c.html#T33" target="_self">Th33</a>, <a class="ref" href="jordan2c.html#T36" target="_self">Th36</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="125_1_1"><b>suppose </b></a><a NAME="E1:125_1_1"/>
<font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 ;<br/><div class="add"><a NAME="E2:125_1_1"/><b>then </b>
<font color="Maroon">B1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K4">Down</a> <font color="Maroon">A1</font>,<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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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> <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>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, , <a class="ref" href="jordan2c.html#T35" target="_self">Th35</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:125_1_1"/><b>then </b>
<font color="Maroon">B1</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">A1</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>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E4:125_1_1"/><b>then </b>
<font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">A1</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>
 <b>by </b><i><a class="ref" href="jordan2c.html#T31" target="_self">Th31</a>, , <a class="ref" href="connsp_1.html#T37">CONNSP_1:37</a></i>;<br/><a NAME="E5:125_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1" target="_self">BDD</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>
 <b>by </b><i><a class="ref" href="jordan2c.html#T27" target="_self">Th27</a>, <a class="ref" href="jordan2c.html#T28" target="_self">Th28</a>, <a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">ep</font> = <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</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/><b>reconsider </b><font color="Maroon">G2</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">ep</font>,<font color="Maroon">a</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="topreal3.html#T13">TOPREAL3:13</a></i>;<br/><a NAME="E8:125_1_1"/>
the <a href="metric_1.html#U1">distance</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> is <a href="metric_1.html#V2">Reflexive</a>
 <b>by </b><i><a class="ref" href="metric_1.html#D7">METRIC_1:def 7</a></i>;<br/><a NAME="E9:125_1_1"/><b>then </b>
the <a href="metric_1.html#U1">distance</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">ep</font>,<font color="Maroon">ep</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="metric_1.html#D3">METRIC_1:def 3</a></i>;<br/><a NAME="E10:125_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">ep</font>,<font color="Maroon">ep</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/><a NAME="E11:125_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">ep</font>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T27" target="_self">Th27</a>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E12:125_1_1"/>
<font color="Maroon">G2</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="goboard6.html#T6">GOBOARD6:6</a></i>;<br/><a NAME="E13:125_1_1"/><b>then </b>
 <a href="jordan2c.html#K1" target="_self">BDD</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="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G2</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T37" target="_self">Th37</a>, <a class="ref" href="jordan2c.html#T38" target="_self">Th38</a>, <a class="ref" href="pre_topc.html#D13">PRE_TOPC:def 13</a></i>;<br/><b>then consider </b><font color="Maroon">t2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:125_1_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">t2</font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K1" target="_self">BDD</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> &amp; <font color="Maroon">t2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>reconsider </b><font color="Maroon">qt2</font> = <font color="Maroon">t2</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><a class="ref" href="jordan2c.html#T39" target="_self">Th39</a></i>;<br/><a NAME="E17:125_1_1"/>
( <font color="Maroon">t2</font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K1" target="_self">BDD</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> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">qt2</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="jordan2c.html#T39" target="_self">Th39</a>, </i>;<br/><b>hence </b><a NAME="E18:125_1_1"/>
 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#R2">in</a>  <a href="jordan2c.html#K1" target="_self">BDD</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> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Olive">q</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="125_1_2"><b>suppose </b></a><a NAME="E1:125_1_2"/>
<font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font>
 ;<br/><div class="add"><a NAME="E2:125_1_2"/><b>then </b>
<font color="Maroon">B1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K4">Down</a> <font color="Maroon">A2</font>,<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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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> <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>by </b><i><a class="ref" href="jordan2c.html#T30" target="_self">Th30</a>, , <a class="ref" href="jordan2c.html#T35" target="_self">Th35</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:125_1_2"/><b>then </b>
<font color="Maroon">B1</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">A2</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>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E4:125_1_2"/><b>then </b>
<font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">A2</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>
 <b>by </b><i><a class="ref" href="jordan2c.html#T31" target="_self">Th31</a>, , <a class="ref" href="connsp_1.html#T37">CONNSP_1:37</a></i>;<br/><a NAME="E5:125_1_2"/><b>then </b><i><font color="Green">E130</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1" target="_self">BDD</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>
 <b>by </b><i><a class="ref" href="jordan2c.html#T27" target="_self">Th27</a>, <a class="ref" href="jordan2c.html#T28" target="_self">Th28</a>, <a class="ref" href="jordan2c.html#T30" target="_self">Th30</a>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">ep</font> = <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</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/><b>reconsider </b><font color="Maroon">G2</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">ep</font>,<font color="Maroon">a</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="topreal3.html#T13">TOPREAL3:13</a></i>;<br/><a NAME="E8:125_1_2"/>
the <a href="metric_1.html#U1">distance</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> is <a href="metric_1.html#V2">Reflexive</a>
 <b>by </b><i><a class="ref" href="metric_1.html#D7">METRIC_1:def 7</a></i>;<br/><a NAME="E9:125_1_2"/><b>then </b>
the <a href="metric_1.html#U1">distance</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">ep</font>,<font color="Maroon">ep</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="metric_1.html#D3">METRIC_1:def 3</a></i>;<br/><a NAME="E10:125_1_2"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">ep</font>,<font color="Maroon">ep</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/><a NAME="E11:125_1_2"/><b>then </b><i><font color="Green">E131</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">ep</font>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T27" target="_self">Th27</a>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E12:125_1_2"/>
<font color="Maroon">G2</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="goboard6.html#T6">GOBOARD6:6</a></i>;<br/><a NAME="E13:125_1_2"/><b>then </b>
 <a href="jordan2c.html#K1" target="_self">BDD</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="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G2</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T37" target="_self">Th37</a>, <a class="ref" href="jordan2c.html#T38" target="_self">Th38</a>, <a class="ref" href="pre_topc.html#D13">PRE_TOPC:def 13</a></i>;<br/><b>then consider </b><font color="Maroon">t2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:125_1_2"/><i><font color="Green">E152</font></i>: 
( <font color="Maroon">t2</font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K1" target="_self">BDD</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> &amp; <font color="Maroon">t2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>reconsider </b><font color="Maroon">qt2</font> = <font color="Maroon">t2</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><a class="ref" href="jordan2c.html#T39" target="_self">Th39</a></i>;<br/><a NAME="E17:125_1_2"/>
( <font color="Maroon">t2</font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K1" target="_self">BDD</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> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">qt2</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="jordan2c.html#T39" target="_self">Th39</a>, </i>;<br/><b>hence </b><a NAME="E18:125_1_2"/>
 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#R2">in</a>  <a href="jordan2c.html#K1" target="_self">BDD</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> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Olive">q</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
