<?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">A</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> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><a NAME="E1:102"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">A</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> <font color="Maroon">n</font></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="hidden.html#R1">=</a> <font color="Maroon">a</font> </span>}</span>  )
 ;<br/>

<a NAME="E2:102"/>
<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> <font color="Maroon">n</font></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">a</font> </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> <font color="Maroon">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_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="E1:102_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> <font color="Maroon">n</font></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">a</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">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> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E3:102_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></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/>
<b>thus </b><a NAME="E4:102_1"/>
<font color="Maroon">x</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> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan2c.html#D3" target="_self">Def3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">W</font> = <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> <font color="Maroon">n</font></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">a</font> </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">W</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> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P</font> = <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> <font color="Maroon">n</font></span>)</span> ;<br/>
<a NAME="E6:102"/><i><font color="Green">E57</font></i>: 
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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</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">P</font>,<span class="p2">(<span class="default"><font color="Maroon">A</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> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E8:102"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_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="E1:102_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">q</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> <font color="Maroon">n</font></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">q1</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> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E4:102_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q1</font></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#T17" target="_self">Th17</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:102_2_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:102_2_1"/><b>then </b>
 ex <font color="Olive">q2</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> <font color="Maroon">n</font></span>)</span> st <br/>( <font color="Olive">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:102_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan2c.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:102_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</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="E9:102"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="toprns_1.html#T24">TOPRNS_1:24</a></i>;<br/>
<a NAME="E10:102"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">G</font> = <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  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> <font color="Maroon">n</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E12:102"/><i><font color="Green">E63</font></i>: 
not <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">G</font> is <a href="struct_0.html#V3">empty</a>
 
;<br/>
<a NAME="E13:102"/><i><font color="Green">E64</font></i>: 
 <a href="connsp_3.html#K4">Down</a> <font color="Maroon">P</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <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="jordan2c.html#T17" target="_self">Th17</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E14:102"/><i><font color="Green">E65</font></i>: 
 <a href="connsp_3.html#K4">Down</a> <font color="Maroon">P</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </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="E15:102"/>
<font color="Maroon">P</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i/>;<br/>
<a NAME="E16:102"/><b>then </b>
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</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="E17:102"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K4">Down</a> <font color="Maroon">P</font>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> is <a href="connsp_1.html#V1">connected</a>
 
<b>by </b><i>, , <a class="ref" href="pre_topc.html#T28">PRE_TOPC:28</a></i>;<br/>
<a NAME="E18:102"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="connsp_3.html#K4">Down</a> <font color="Maroon">P</font>,<span class="p1">(<span class="default"><font color="Maroon">A</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="connsp_1.html#D3">CONNSP_1:def 3</a></i>;<br/>
<a NAME="E19:102"/><b>then </b><i><font color="Green">E67</font></i>: 
 <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">P</font>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#D4" target="_self">Def4</a>, <a class="ref" href="jordan2c.html#T18" target="_self">Th18</a>, <a class="ref" href="connsp_3.html#T9">CONNSP_3:9</a></i>;<br/>
<a NAME="E20:102"/><b>then </b><i><font color="Green">E130</font></i>: 
 <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">P</font>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> 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="E21:102"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">P</font> <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">P</font>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T17" target="_self">Th17</a>, <a class="ref" href="jordan2c.html#D4" target="_self">Def4</a>, , , <a class="ref" href="connsp_3.html#T13">CONNSP_3:13</a></i>;<br/>
<a NAME="E22:102"/>
 <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">P</font>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> is  <a href="tbsp_1.html#V6">bounded</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_3"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">D2</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">P</font>,<span class="p2">(<span class="default"><font color="Maroon">A</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> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="jordan2c.html#D3" target="_self">Def3</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">D2</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">D</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:102_3_1"/>
not <font color="Maroon">D2</font> is <a href="jordan2c.html#V1" target="_self">Bounded</a>
 ;<br/><b>then consider </b><font color="Maroon">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> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E3:102_3_1"/><i><font color="Green">E152</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">p</font> =  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font>;<br/><a NAME="E4:102_3_1"/><i><font color="Green">E154</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</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="toprns_1.html#T24">TOPRNS_1:24</a></i>;<br/><a NAME="E5:102_3_1"/><i><font color="Green">E157</font></i>: 
 <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T22" target="_self">Th22</a>, <a class="ref" href="toprns_1.html#T24">TOPRNS_1:24</a></i>;<br/><a NAME="E6:102_3_1"/>
<font color="Maroon">D</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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E7:102_3_1"/><b>then </b><i><font color="Green">E158</font></i>: 
<font color="Maroon">D2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</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/><a NAME="E8:102_3_1"/><b>then </b><i><font color="Green">E159</font></i>: 
<font color="Maroon">D2</font> <a href="hidden.html#R1">=</a>  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">D2</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  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> <font color="Maroon">n</font></span>)</span> <b>by </b><i>, <a class="ref" href="jordan2c.html#T17" target="_self">Th17</a></i>;<br/><b>reconsider </b><font color="Maroon">RR</font> = <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font> as  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a> ;<br/><a NAME="E11:102_3_1"/>
<font color="Maroon">RR</font> is <a href="connsp_2.html#V1">locally_connected</a>
 <b>by </b><i>, </i>;<br/><a NAME="E12:102_3_1"/><b>then </b>
 <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">P</font>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="jordan2c.html#T21" target="_self">Th21</a>, <a class="ref" href="connsp_2.html#T21">CONNSP_2:21</a></i>;<br/><b>then consider </b><font color="Maroon">G</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> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E14:102_3_1"/><i><font color="Green">E160</font></i>: 
( <font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">D2</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/><a NAME="E15:102_3_1"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font>
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T25" target="_self">Th25</a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E16:102_3_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, </i>;<br/><a NAME="E17:102_3_1"/><b>then </b>
( <font color="Maroon">D2</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">D2</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T25" target="_self">Th25</a>, <a class="ref" href="jordan2c.html#T26" target="_self">Th26</a>, <a class="ref" href="connsp_1.html#T24">CONNSP_1:24</a>, <a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">f1</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E19:102_3_1"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">f1</font> is <a href="pre_topc.html#V5">continuous</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font> &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="jordan2c.html#T17" target="_self">Th17</a>, , <a class="ref" href="jordan2c.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan2c.html#T24" target="_self">Th24</a>, <a class="ref" href="jordan2c.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E20:102_3_1"/><i><font color="Green">E163</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><a href="jordan2c.html#K6" target="_self">pi</a> <font color="Maroon">f1</font>,0</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#T23" target="_self">Th23</a>, , <a class="ref" href="jordan2c.html#D2" target="_self">Def2</a>, <a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a></i>;<br/><a NAME="E21:102_3_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><a href="jordan2c.html#K6" target="_self">pi</a> <font color="Maroon">f1</font>,1</span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T22" target="_self">Th22</a>, , <a class="ref" href="jordan2c.html#D2" target="_self">Def2</a>, <a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a></i>;<br/><b>then consider </b><font color="Maroon">t0</font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E23:102_3_1"/><i><font color="Green">E164</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><a href="jordan2c.html#K6" target="_self">pi</a> <font color="Maroon">f1</font>,<font color="Maroon">t0</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T27" target="_self">Th27</a>, </i>;<br/><b>reconsider </b><font color="Maroon">q2</font> = <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t0</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> <font color="Maroon">n</font></span>)</span> ;<br/><a NAME="E25:102_3_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T28" target="_self">Th28</a>, <a class="ref" href="jordan2c.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E26:102_3_1"/><b>then </b><i><font color="Green">E165</font></i>: 
<font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><a NAME="E27:102_3_1"/>
<font color="Maroon">t0</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
 ;<br/><a NAME="E28:102_3_1"/><b>then </b>
<font color="Maroon">t0</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E29:102_3_1"/><b>then </b>
<font color="Maroon">q2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E30:102_3_1"/><b>then </b>
<font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font>
 <b>by </b><i/>;<br/><a NAME="E31:102_3_1"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="subset_1.html#K1">{}</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> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E32:102_3_1"/><b>then </b>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E33:102_3_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:102_3"/>
 <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">P</font>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> is  <a href="tbsp_1.html#V6">bounded</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E23:102"/><b>then </b><i><font color="Green">E166</font></i>: 
<font color="Maroon">P1</font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T21" target="_self">Th21</a>, </i>;<br/>
<a NAME="E24:102"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">P1</font> <a href="tarski.html#R1">c=</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> <font color="Maroon">n</font></span>)</span> : <font color="Olive">B</font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">A</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_4"><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="E1:102_4"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 ;<br/>

<a NAME="E2:102_4"/>
<font color="Maroon">P1</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> <font color="Maroon">n</font></span>)</span> : <font color="Olive">B</font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">A</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="jordan2c.html#T22" target="_self">Th22</a></i>;<br/>
<b>hence </b><a NAME="E3:102_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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> <font color="Maroon">n</font></span>)</span> : <font color="Olive">B</font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">A</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan2c.html#T24" target="_self">Th24</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_5"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:102_5_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_5_1_1"><b>case </b></a><a NAME="E1:102_5_1_1"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 ;<br/><div class="add"><a NAME="E2:102_5_1_1"/>
 <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> <font color="Maroon">n</font></span>)</span> : <font color="Olive">B</font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">A</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_5_1_1_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="E1:102_5_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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> <font color="Maroon">n</font></span>)</span> : <font color="Olive">B</font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">A</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:102_5_1_1_1"/><i><font color="Green">E170</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</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> <font color="Maroon">n</font></span>)</span> : <font color="Olive">B</font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">A</font> </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> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E5:102_5_1_1_1"/><i><font color="Green">E171</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">B</font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/>
<a NAME="E6:102_5_1_1_1"/><i><font color="Green">E172</font></i>: 
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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</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>consider </b><font color="Maroon">C</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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E8:102_5_1_1_1"/><i><font color="Green">E205</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">C</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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Maroon">C</font> is  <a href="tbsp_1.html#V6">bounded</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <b>by </b><i>, , <a class="ref" href="jordan2c.html#T26" target="_self">Th26</a></i>;<br/>
<b>reconsider </b><font color="Maroon">E</font> = <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  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> <font color="Maroon">n</font></span>)</span> <b>by </b><i>, <a class="ref" href="jordan2c.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E11:102_5_1_1_1"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">E</font></span>)</span>
 
;<br/>
<a NAME="E12:102_5_1_1_1"/><b>then </b><i><font color="Green">E207</font></i>: 
( <font color="Maroon">p</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> <font color="Maroon">n</font></span>)</span> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a>, <a class="ref" href="jordan2c.html#T25" target="_self">Th25</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q2</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#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="jordan2c.html#T25" target="_self">Th25</a>, </i>;<br/>
<a NAME="E14:102_5_1_1_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="jordan2c.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E15:102_5_1_1_1"/><b>then </b><i><font color="Green">E208</font></i>: 
( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> or <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_5_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_5_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:102_5_1_1_1_1_1"/>
( <font color="Maroon">p</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> <font color="Maroon">n</font></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#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> </span>}</span>  or <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q1</font> where <font color="Olive">q1</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> <font color="Maroon">n</font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> </span>}</span>  )
 </span><b>by </b><i><a class="ref" href="jordan2c.html#T28" target="_self">Th28</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_5_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:102_5_1_1_1_1_1_1"/><i><font color="Green">E209</font></i>: 
<font color="Maroon">p</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> <font color="Maroon">n</font></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#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:102_5_1_1_1_1_1_1"/>
<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> <font color="Maroon">n</font></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#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_5_1_1_1_1_1_1_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="E1:102_5_1_1_1_1_1_1_1"/>
<font color="Maroon">z</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> <font color="Maroon">n</font></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#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">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> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E3:102_5_1_1_1_1_1_1_1"/><i><font color="Green">E210</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_5_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:102_5_1_1_1_1_1_1_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:102_5_1_1_1_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive">q2</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> <font color="Maroon">n</font></span>)</span> st <br/>( <font color="Olive">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:102_5_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan2c.html#T30" target="_self">Th30</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:102_5_1_1_1_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><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/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">Q</font> = <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> <font color="Maroon">n</font></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#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> </span>}</span>  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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><b>reconsider </b><font color="Maroon">Q</font> = <font color="Maroon">Q</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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> ;<br/><a NAME="E5:102_5_1_1_1_1_1_1"/>
<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> <font color="Maroon">n</font></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#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> </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> <font color="Maroon">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_5_1_1_1_1_1_1_2"><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="E1:102_5_1_1_1_1_1_1_2"/>
<font color="Maroon">z</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> <font color="Maroon">n</font></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#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">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> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E3:102_5_1_1_1_1_1_1_2"/><i><font color="Green">E211</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> )
 ;<br/>
<b>thus </b><a NAME="E4:102_5_1_1_1_1_1_1_2"/>
<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> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T30" target="_self">Th30</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">P2</font> = <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> <font color="Maroon">n</font></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#NR4" target="_self">&gt;</a> <font color="Maroon">a</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> <font color="Maroon">n</font></span>)</span> ;<br/><a NAME="E7:102_5_1_1_1_1_1_1"/>
<font color="Maroon">P2</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="jordan2c.html#T24" target="_self">Th24</a>, <a class="ref" href="jordan2c.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E8:102_5_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E212</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P2</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="E9:102_5_1_1_1_1_1_1"/>
<font color="Maroon">P2</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/><b>then reconsider </b><font color="Maroon">W2</font> = <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> <font color="Maroon">n</font></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#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> ;<br/><a NAME="E11:102_5_1_1_1_1_1_1"/><i><font color="Green">E213</font></i>: 
<font color="Maroon">P2</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="jordan2c.html#T24" target="_self">Th24</a>, <a class="ref" href="jordan2c.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E12:102_5_1_1_1_1_1_1"/>
<font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font>
 ;<br/><a NAME="E13:102_5_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E214</font></i>: 
not <font color="Maroon">W2</font> is <a href="tbsp_1.html#V6">bounded</a>
 <b>by </b><i><a class="ref" href="jordan2c.html#T24" target="_self">Th24</a>, <a class="txt" href="jordan2c.html#E13:102"><i><font color="Green">E64</font></i></a></i>;<br/><div><i><font color="Green">E215</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="102_5_1_1_1_1_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:102_5_1_1_1_1_1_1_3"/>
<font color="Maroon">W2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font>
 ;<br/><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:102_5_1_1_1_1_1_1_3"/><i><font color="Green">E216</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:102_5_1_1_1_1_1_1_3"/><i><font color="Green">E217</font></i>: 
 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> <font color="Maroon">n</font></span>)</span> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</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#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="jordan2c.html#T35" target="_self">Th35</a></i>;<br/><a NAME="E5:102_5_1_1_1_1_1_1_3"/>
 ex <font color="Olive">q2</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> <font color="Maroon">n</font></span>)</span> st <br/>( <font color="Olive">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T35" target="_self">Th35</a></i>;<br/><b>hence </b><a NAME="E6:102_5_1_1_1_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan2c.html#T36" target="_self">Th36</a></i>;<br/></div><b>end;</b></div><a NAME="E15:102_5_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">W2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> </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="E16:102_5_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">P2</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T32">SUBSET_1:32</a></i>;<br/><a NAME="E17:102_5_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E218</font></i>: 
<font color="Maroon">W2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><a NAME="E18:102_5_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E219</font></i>: 
<font color="Maroon">P2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E19:102_5_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T35" target="_self">Th35</a>, <a class="ref" href="pre_topc.html#T28">PRE_TOPC:28</a></i>;<br/><a NAME="E20:102_5_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E220</font></i>: 
<font color="Maroon">Q</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="jordan2c.html#T30" target="_self">Th30</a>, <a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a></i>;<br/><a NAME="E21:102_5_1_1_1_1_1_1"/>
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">P2</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T36" target="_self">Th36</a></i>;<br/><a NAME="E22:102_5_1_1_1_1_1_1"/><b>then </b>
 <a href="connsp_3.html#K5">Up</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K1">Component_of</a> <font color="Maroon">Q</font></span>)</span> <a href="jordan2c.html#R2" target="_self">is_outside_component_of</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T31" target="_self">Th31</a>, , <a class="ref" href="jordan2c.html#T33" target="_self">Th33</a>, <a class="txt" href="jordan2c.html#E18:102"><i><font color="Green">E66</font></i></a></i>;<br/><a NAME="E23:102_5_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E221</font></i>: 
 <a href="connsp_3.html#K1">Component_of</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><a NAME="E24:102_5_1_1_1_1_1_1"/>
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="connsp_3.html#K1">Component_of</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T37" target="_self">Th37</a>, <a class="ref" href="connsp_3.html#T1">CONNSP_3:1</a></i>;<br/><a NAME="E25:102_5_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E222</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="connsp_3.html#K1">Component_of</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a></i>;<br/><a NAME="E26:102_5_1_1_1_1_1_1"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/><a NAME="E27:102_5_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan2c.html#T38" target="_self">Th38</a>, <a class="ref" href="jordan2c.html#T39" target="_self">Th39</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E28:102_5_1_1_1_1_1_1"/><b>then </b>
 <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/><b>hence </b><a NAME="E29:102_5_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_5_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:102_5_1_1_1_1_1_2"/><i><font color="Green">E223</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q1</font> where <font color="Olive">q1</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> <font color="Maroon">n</font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:102_5_1_1_1_1_1_2"/>
 <a href="connsp_3.html#K4">Down</a> <font color="Maroon">P</font>,<span class="p1">(<span class="default"><font color="Maroon">A</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">P</font>,<span class="p2">(<span class="default"><font color="Maroon">A</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/><b>hence </b><a NAME="E3:102_5_1_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T29" target="_self">Th29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E17:102_5_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E3:102_5_1_1"/><b>then </b>
<font color="Maroon">P1</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> <font color="Maroon">n</font></span>)</span> : <font color="Olive">B</font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">A</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#D10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E4:102_5_1_1"/>
 ex <font color="Olive">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> <font color="Maroon">n</font></span>)</span> st <br/>( <font color="Olive">B</font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">A</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R1">=</a>  <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="jordan2c.html#T22" target="_self">Th22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_5_1_2"><b>case </b></a><a NAME="E1:102_5_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2
 ;<br/><div class="add"><a NAME="E2:102_5_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K23">+</a> 1
 ;<br/><a NAME="E3:102_5_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:102_5_1_2"/><b>then </b><i><font color="Green">E224</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:102_5_1_2"/>
 <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> <font color="Maroon">n</font></span>)</span> : <font color="Olive">B</font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">A</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_5_1_2_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="E1:102_5_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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> <font color="Maroon">n</font></span>)</span> : <font color="Olive">B</font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">A</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:102_5_1_2_1"/><i><font color="Green">E225</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</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> <font color="Maroon">n</font></span>)</span> : <font color="Olive">B</font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">A</font> </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> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E5:102_5_1_2_1"/><i><font color="Green">E232</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">B</font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/>
<a NAME="E6:102_5_1_2_1"/><i><font color="Green">E322</font></i>: 
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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</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>consider </b><font color="Maroon">C</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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E8:102_5_1_2_1"/><i><font color="Green">E323</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">C</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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Maroon">C</font> is  <a href="tbsp_1.html#V6">bounded</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <b>by </b><i>, , <a class="ref" href="jordan2c.html#T26" target="_self">Th26</a></i>;<br/>
<b>reconsider </b><font color="Maroon">E</font> = <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  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> <font color="Maroon">n</font></span>)</span> <b>by </b><i>, <a class="ref" href="jordan2c.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E11:102_5_1_2_1"/><i><font color="Green">E324</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">E</font></span>)</span>
 
;<br/>
<a NAME="E12:102_5_1_2_1"/><b>then </b>
<font color="Maroon">p</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> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E13:102_5_1_2_1"/><b>then </b><i><font color="Green">E325</font></i>: 
( <font color="Maroon">p</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> <font color="Maroon">n</font></span>)</span> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</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">p</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> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="jordan2c.html#T25" target="_self">Th25</a>, </i>;<br/>
<a NAME="E15:102_5_1_2_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="jordan2c.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E16:102_5_1_2_1"/><b>then </b><i><font color="Green">E326</font></i>: 
( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> or <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_5_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_5_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:102_5_1_2_1_1_1"/>
( <font color="Maroon">p</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> <font color="Maroon">n</font></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#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> </span>}</span>  or <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q1</font> where <font color="Olive">q1</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> <font color="Maroon">n</font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> </span>}</span>  )
 </span><b>by </b><i><a class="ref" href="jordan2c.html#T28" target="_self">Th28</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_5_1_2_1_1_1_1"><b>case </b></a><a NAME="E1:102_5_1_2_1_1_1_1"/>
<font color="Maroon">p</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> <font color="Maroon">n</font></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#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">q0</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> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E3:102_5_1_2_1_1_1_1"/><i><font color="Green">E327</font></i>: 
( <font color="Maroon">q0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q0</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> )
 ;<br/><a NAME="E4:102_5_1_2_1_1_1_1"/>
<font color="Maroon">q0</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/><b>then consider </b><font color="Maroon">r0</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:102_5_1_2_1_1_1_1"/><i><font color="Green">E328</font></i>: 
<font color="Maroon">q0</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">r0</font></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T24" target="_self">Th24</a>, <a class="ref" href="finseq_2.html#T117">FINSEQ_2:117</a></i>;<br/><a NAME="E7:102_5_1_2_1_1_1_1"/><i><font color="Green">E329</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q0</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">r0</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T30" target="_self">Th30</a>, </i>;<br/><div><i><font color="Green">E330</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="102_5_1_2_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_5_1_2_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:102_5_1_2_1_1_1_1_1_1"/>
( <font color="Maroon">r0</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">r0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_5_1_2_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:102_5_1_2_1_1_1_1_1_1_1"/>
<font color="Maroon">r0</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:102_5_1_2_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">r0</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">r0</font>
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E3:102_5_1_2_1_1_1_1_1_1_1"/>
( <font color="Maroon">p</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> <font color="Maroon">n</font></span>)</span> :  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">r</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> ) </span>}</span>  or <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q1</font> where <font color="Olive">q1</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> <font color="Maroon">n</font></span>)</span> :  ex <font color="Olive">r1</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">q1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">r1</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Olive">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">a</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, <a class="ref" href="jordan2c.html#T30" target="_self">Th30</a>, <a class="ref" href="jordan2c.html#T31" target="_self">Th31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_5_1_2_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:102_5_1_2_1_1_1_1_1_1_2"/>
<font color="Maroon">r0</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:102_5_1_2_1_1_1_1_1_1_2"/><b>then </b>
 <a href="binop_2.html#K7">-</a> <font color="Maroon">r0</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, <a class="ref" href="jordan2c.html#T31" target="_self">Th31</a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E3:102_5_1_2_1_1_1_1_1_1_2"/><b>then </b>
 <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">r0</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><b>hence </b><a NAME="E4:102_5_1_2_1_1_1_1_1_1_2"/>
( <font color="Maroon">p</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> <font color="Maroon">n</font></span>)</span> :  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">r</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> ) </span>}</span>  or <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q1</font> where <font color="Olive">q1</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> <font color="Maroon">n</font></span>)</span> :  ex <font color="Olive">r1</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">q1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">r1</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Olive">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">a</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, <a class="ref" href="jordan2c.html#T30" target="_self">Th30</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_5_1_2_1_1_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_5_1_2_1_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:102_5_1_2_1_1_1_1_2_1"/>
( <font color="Maroon">p</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> <font color="Maroon">n</font></span>)</span> :  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">r</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> ) </span>}</span>  or <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q1</font> where <font color="Olive">q1</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> <font color="Maroon">n</font></span>)</span> :  ex <font color="Olive">r1</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">q1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">r1</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Olive">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">a</font> ) </span>}</span>  )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_5_1_2_1_1_1_1_2_1_1"><b>case </b></a><a NAME="E1:102_5_1_2_1_1_1_1_2_1_1"/><i><font color="Green">E331</font></i>: 
<font color="Maroon">p</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> <font color="Maroon">n</font></span>)</span> :  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">r</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> ) </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:102_5_1_2_1_1_1_1_2_1_1"/>
<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> <font color="Maroon">n</font></span>)</span> :  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">r</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_5_1_2_1_1_1_1_2_1_1_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="E1:102_5_1_2_1_1_1_1_2_1_1_1"/>
<font color="Maroon">z</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> <font color="Maroon">n</font></span>)</span> :  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">r</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">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> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E3:102_5_1_2_1_1_1_1_2_1_1_1"/><i><font color="Green">E332</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp;  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">r</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:102_5_1_2_1_1_1_1_2_1_1_1"/><i><font color="Green">E333</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="jordan2c.html#T35" target="_self">Th35</a></i>;<br/>
<a NAME="E6:102_5_1_2_1_1_1_1_2_1_1_1"/><i><font color="Green">E334</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="jordan2c.html#T36" target="_self">Th36</a></i>;<br/>
<b>reconsider </b><font color="Maroon">xr</font> = <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="binarith.html#K11">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="jordan2c.html#T24" target="_self">Th24</a></i>;<br/>
<i><font color="Green">E335</font></i>: <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> = 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">xr</font></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="jordan2c.html#T36" target="_self">Th36</a>, <a class="ref" href="jgraph_1.html#D5">JGRAPH_1:def 5</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">xr</font></span>)</span></span>)</span>

;<br/>
<a NAME="E9:102_5_1_2_1_1_1_1_2_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">xr</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E10:102_5_1_2_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">xr</font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">xr</font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i><a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E11:102_5_1_2_1_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E336</font></i>: 
 <a href="jordan2b.html#K1">Proj</a> <font color="Maroon">q</font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">xr</font> <a href="goboard1.html#K1">.</a> 1
 
<b>by </b><i><a class="ref" href="jordan2c.html#T36" target="_self">Th36</a>, <a class="ref" href="jordan2b.html#D1">JORDAN2B:def 1</a></i>;<br/>
<a NAME="E12:102_5_1_2_1_1_1_1_2_1_1_1"/><i><font color="Green">E337</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">xr</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="jordan2c.html#T24" target="_self">Th24</a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E13:102_5_1_2_1_1_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">xr</font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">q</font>,1</span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="jordan2c.html#T39" target="_self">Th39</a>, <a class="ref" href="rvsum_1.html#T79">RVSUM_1:79</a></i>;<br/>
<a NAME="E14:102_5_1_2_1_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E338</font></i>: 
 <a href="euclid.html#K11">sqr</a> <font color="Maroon">xr</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">q</font>,1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">q</font>,1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">q</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T158">COMPLEX1:158</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="jordan2c.html#T39" target="_self">Th39</a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/>
<b>then </b><i><font color="Green">E339</font></i>: <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> = 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="jordan2c.html#T38" target="_self">Th38</a>, , <a class="ref" href="finsop_1.html#T12">FINSOP_1:12</a></i>
<br/>.= 
<font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="jordan2c.html#T37" target="_self">Th37</a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_5_1_2_1_1_1_1_2_1_1_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:102_5_1_2_1_1_1_1_2_1_1_1_4"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:102_5_1_2_1_1_1_1_2_1_1_1_4"/><b>then </b>
 ex <font color="Olive">q2</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> <font color="Maroon">n</font></span>)</span> st <br/>( <font color="Olive">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:102_5_1_2_1_1_1_1_2_1_1_1_4"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan2c.html#T36" target="_self">Th36</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E18:102_5_1_2_1_1_1_1_2_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="jordan2c.html#T35" target="_self">Th35</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">Q</font> = <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> <font color="Maroon">n</font></span>)</span> :  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">r</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> ) </span>}</span>  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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><b>reconsider </b><font color="Maroon">Q</font> = <font color="Maroon">Q</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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> ;<br/><a NAME="E5:102_5_1_2_1_1_1_1_2_1_1"/>
<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> <font color="Maroon">n</font></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#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> </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> <font color="Maroon">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_5_1_2_1_1_1_1_2_1_1_2"><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="E1:102_5_1_2_1_1_1_1_2_1_1_2"/>
<font color="Maroon">z</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> <font color="Maroon">n</font></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#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">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> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E3:102_5_1_2_1_1_1_1_2_1_1_2"/><i><font color="Green">E340</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> )
 ;<br/>
<b>thus </b><a NAME="E4:102_5_1_2_1_1_1_1_2_1_1_2"/>
<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> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T35" target="_self">Th35</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">P2</font> = <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> <font color="Maroon">n</font></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#NR4" target="_self">&gt;</a> <font color="Maroon">a</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> <font color="Maroon">n</font></span>)</span> ;<br/><a NAME="E7:102_5_1_2_1_1_1_1_2_1_1"/>
<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> <font color="Maroon">n</font></span>)</span> :  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">r</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> ) </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> <font color="Maroon">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_5_1_2_1_1_1_1_2_1_1_3"><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="E1:102_5_1_2_1_1_1_1_2_1_1_3"/>
<font color="Maroon">z</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> <font color="Maroon">n</font></span>)</span> :  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">r</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">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> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E3:102_5_1_2_1_1_1_1_2_1_1_3"/><i><font color="Green">E341</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp;  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">r</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> ) )
 ;<br/>
<b>thus </b><a NAME="E4:102_5_1_2_1_1_1_1_2_1_1_3"/>
<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> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T35" target="_self">Th35</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">P3</font> = <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> <font color="Maroon">n</font></span>)</span> :  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">r</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</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> <font color="Maroon">n</font></span>)</span> ;<br/><a NAME="E9:102_5_1_2_1_1_1_1_2_1_1"/>
<font color="Maroon">P2</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/><b>then reconsider </b><font color="Maroon">W2</font> = <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> <font color="Maroon">n</font></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#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> ;<br/><b>reconsider </b><font color="Maroon">W3</font> = <font color="Maroon">P3</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/><a NAME="E12:102_5_1_2_1_1_1_1_2_1_1"/>
<font color="Maroon">W3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P3</font>
 ;<br/><a NAME="E13:102_5_1_2_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E344</font></i>: 
<font color="Maroon">P3</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="jordan2c.html#T24" target="_self">Th24</a>, </i>;<br/><a NAME="E14:102_5_1_2_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E345</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P3</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="E15:102_5_1_2_1_1_1_1_2_1_1"/><i><font color="Green">E346</font></i>: 
not <font color="Maroon">W3</font> is <a href="tbsp_1.html#V6">bounded</a>
 <b>by </b><i><a class="ref" href="jordan2c.html#T24" target="_self">Th24</a>, </i>;<br/><div><i><font color="Green">E347</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="102_5_1_2_1_1_1_1_2_1_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:102_5_1_2_1_1_1_1_2_1_1_4"/><i><font color="Green">E348</font></i>: 
not <font color="Maroon">W2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">W2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>;<br/><a NAME="E3:102_5_1_2_1_1_1_1_2_1_1_4"/><i><font color="Green">E349</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="jordan2c.html#T39" target="_self">Th39</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:102_5_1_2_1_1_1_1_2_1_1_4"/><b>then </b><i><font color="Green">E350</font></i>: 
 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> <font color="Maroon">n</font></span>)</span> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</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#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> )
 ;<br/><a NAME="E5:102_5_1_2_1_1_1_1_2_1_1_4"/>
 ex <font color="Olive">q2</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> <font color="Maroon">n</font></span>)</span> st <br/>( <font color="Olive">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E6:102_5_1_2_1_1_1_1_2_1_1_4"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E17:102_5_1_2_1_1_1_1_2_1_1"/>
<font color="Maroon">W3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_5_1_2_1_1_1_1_2_1_1_5"><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="E1:102_5_1_2_1_1_1_1_2_1_1_5"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W3</font>
 ;<br/>

<b>then consider </b><font color="Maroon">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> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E3:102_5_1_2_1_1_1_1_2_1_1_5"/><i><font color="Green">E351</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp;  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">r</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:102_5_1_2_1_1_1_1_2_1_1_5"/><i><font color="Green">E352</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="jordan2c.html#T39" target="_self">Th39</a></i>;<br/>
<b>reconsider </b><font color="Maroon">xr</font> = <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="binarith.html#K11">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="jordan2c.html#T24" target="_self">Th24</a></i>;<br/>
<i><font color="Green">E353</font></i>: <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> = 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">xr</font></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i>, <a class="ref" href="jgraph_1.html#D5">JGRAPH_1:def 5</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">xr</font></span>)</span></span>)</span>

;<br/>
<a NAME="E8:102_5_1_2_1_1_1_1_2_1_1_5"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">xr</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E9:102_5_1_2_1_1_1_1_2_1_1_5"/><b>then </b>
<font color="Maroon">xr</font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">xr</font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i><a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E10:102_5_1_2_1_1_1_1_2_1_1_5"/><b>then </b><i><font color="Green">E354</font></i>: 
 <a href="jordan2b.html#K1">Proj</a> <font color="Maroon">q</font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">xr</font> <a href="goboard1.html#K1">.</a> 1
 
<b>by </b><i>, <a class="ref" href="jordan2b.html#D1">JORDAN2B:def 1</a></i>;<br/>
<a NAME="E11:102_5_1_2_1_1_1_1_2_1_1_5"/><i><font color="Green">E355</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">xr</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="jordan2c.html#T24" target="_self">Th24</a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E12:102_5_1_2_1_1_1_1_2_1_1_5"/>
<span class="p1">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">xr</font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">q</font>,1</span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T79">RVSUM_1:79</a></i>;<br/>
<a NAME="E13:102_5_1_2_1_1_1_1_2_1_1_5"/><b>then </b><i><font color="Green">E356</font></i>: 
 <a href="euclid.html#K11">sqr</a> <font color="Maroon">xr</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">q</font>,1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T40" target="_self">Th40</a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">q</font>,1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">q</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T158">COMPLEX1:158</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/>
<a NAME="E15:102_5_1_2_1_1_1_1_2_1_1_5"/><b>then </b><i><font color="Green">E357</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font>
 
<b>by </b><i>, , <a class="ref" href="finsop_1.html#T12">FINSOP_1:12</a></i>;<br/>
<a NAME="E16:102_5_1_2_1_1_1_1_2_1_1_5"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a>, , </i>;<br/>
<b>hence </b><a NAME="E17:102_5_1_2_1_1_1_1_2_1_1_5"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T39" target="_self">Th39</a>, , </i>;<br/>


</div><b>end;</b></div><a NAME="E18:102_5_1_2_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E358</font></i>: 
<font color="Maroon">W3</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E19:102_5_1_2_1_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">W3</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="jordan2c.html#T38" target="_self">Th38</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><a NAME="E20:102_5_1_2_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E359</font></i>: 
<font color="Maroon">W3</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E21:102_5_1_2_1_1_1_1_2_1_1"/>
<font color="Maroon">W3</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="jordan2c.html#T38" target="_self">Th38</a>, <a class="ref" href="jordan2c.html#T39" target="_self">Th39</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><a NAME="E22:102_5_1_2_1_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">W3</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T32">SUBSET_1:32</a></i>;<br/><a NAME="E23:102_5_1_2_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E360</font></i>: 
<font color="Maroon">W3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><a NAME="E24:102_5_1_2_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E361</font></i>: 
<font color="Maroon">P3</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P3</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E25:102_5_1_2_1_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Q</font>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#T28">PRE_TOPC:28</a></i>;<br/><a NAME="E26:102_5_1_2_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E362</font></i>: 
<font color="Maroon">Q</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="jordan2c.html#T36" target="_self">Th36</a>, <a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a></i>;<br/><a NAME="E27:102_5_1_2_1_1_1_1_2_1_1"/>
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">P3</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i/>;<br/><a NAME="E28:102_5_1_2_1_1_1_1_2_1_1"/><b>then </b>
 <a href="connsp_3.html#K5">Up</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K1">Component_of</a> <font color="Maroon">Q</font></span>)</span> <a href="jordan2c.html#R2" target="_self">is_outside_component_of</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T35" target="_self">Th35</a>, <a class="ref" href="jordan2c.html#T37" target="_self">Th37</a>, , <a class="txt" href="jordan2c.html#E18:102"><i><font color="Green">E66</font></i></a></i>;<br/><a NAME="E29:102_5_1_2_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E363</font></i>: 
 <a href="connsp_3.html#K1">Component_of</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><a NAME="E30:102_5_1_2_1_1_1_1_2_1_1"/>
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="connsp_3.html#K1">Component_of</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T40" target="_self">Th40</a>, <a class="ref" href="connsp_3.html#T1">CONNSP_3:1</a></i>;<br/><a NAME="E31:102_5_1_2_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E364</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="connsp_3.html#K1">Component_of</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T33" target="_self">Th33</a></i>;<br/><a NAME="E32:102_5_1_2_1_1_1_1_2_1_1"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/><a NAME="E33:102_5_1_2_1_1_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">A</font></span>)</span> <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="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E34:102_5_1_2_1_1_1_1_2_1_1"/><b>then </b>
 <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E35:102_5_1_2_1_1_1_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_5_1_2_1_1_1_1_2_1_2"><b>case </b></a><a NAME="E1:102_5_1_2_1_1_1_1_2_1_2"/><i><font color="Green">E365</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q1</font> where <font color="Olive">q1</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> <font color="Maroon">n</font></span>)</span> :  ex <font color="Olive">r1</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">q1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">r1</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Olive">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">a</font> ) </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:102_5_1_2_1_1_1_1_2_1_2"/>
<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> <font color="Maroon">n</font></span>)</span> :  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">r</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">a</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_5_1_2_1_1_1_1_2_1_2_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="E1:102_5_1_2_1_1_1_1_2_1_2_1"/>
<font color="Maroon">z</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> <font color="Maroon">n</font></span>)</span> :  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">r</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">a</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">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> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E3:102_5_1_2_1_1_1_1_2_1_2_1"/><i><font color="Green">E366</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp;  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">r</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">a</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:102_5_1_2_1_1_1_1_2_1_2_1"/><i><font color="Green">E367</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="jordan2c.html#T35" target="_self">Th35</a></i>;<br/>
<a NAME="E6:102_5_1_2_1_1_1_1_2_1_2_1"/><i><font color="Green">E368</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> 0
 
<b>by </b><i>, <a class="ref" href="jordan2c.html#T36" target="_self">Th36</a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<b>reconsider </b><font color="Maroon">xr</font> = <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="binarith.html#K11">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="jordan2c.html#T24" target="_self">Th24</a></i>;<br/>
<i><font color="Green">E369</font></i>: <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> = 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">xr</font></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="jordan2c.html#T36" target="_self">Th36</a>, <a class="ref" href="jgraph_1.html#D5">JGRAPH_1:def 5</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">xr</font></span>)</span></span>)</span>

;<br/>
<a NAME="E9:102_5_1_2_1_1_1_1_2_1_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">xr</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E10:102_5_1_2_1_1_1_1_2_1_2_1"/><b>then </b>
<font color="Maroon">xr</font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">xr</font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i><a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E11:102_5_1_2_1_1_1_1_2_1_2_1"/><b>then </b><i><font color="Green">E370</font></i>: 
 <a href="jordan2b.html#K1">Proj</a> <font color="Maroon">q</font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">xr</font> <a href="goboard1.html#K1">.</a> 1
 
<b>by </b><i><a class="ref" href="jordan2c.html#T36" target="_self">Th36</a>, <a class="ref" href="jordan2b.html#D1">JORDAN2B:def 1</a></i>;<br/>
<a NAME="E12:102_5_1_2_1_1_1_1_2_1_2_1"/><i><font color="Green">E371</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">xr</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="jordan2c.html#T24" target="_self">Th24</a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E13:102_5_1_2_1_1_1_1_2_1_2_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">xr</font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">q</font>,1</span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="jordan2c.html#T39" target="_self">Th39</a>, <a class="ref" href="rvsum_1.html#T79">RVSUM_1:79</a></i>;<br/>
<a NAME="E14:102_5_1_2_1_1_1_1_2_1_2_1"/><b>then </b><i><font color="Green">E372</font></i>: 
 <a href="euclid.html#K11">sqr</a> <font color="Maroon">xr</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">q</font>,1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">q</font>,1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">q</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T158">COMPLEX1:158</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="jordan2c.html#T39" target="_self">Th39</a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/>
<b>then </b><i><font color="Green">E373</font></i>: <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> = 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="jordan2c.html#T38" target="_self">Th38</a>, , <a class="ref" href="finsop_1.html#T12">FINSOP_1:12</a></i>
<br/>.= 
 <a href="binop_2.html#K7">-</a> <font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="jordan2c.html#T37" target="_self">Th37</a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_5_1_2_1_1_1_1_2_1_2_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:102_5_1_2_1_1_1_1_2_1_2_1_4"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:102_5_1_2_1_1_1_1_2_1_2_1_4"/><b>then </b>
 ex <font color="Olive">q2</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> <font color="Maroon">n</font></span>)</span> st <br/>( <font color="Olive">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:102_5_1_2_1_1_1_1_2_1_2_1_4"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan2c.html#T36" target="_self">Th36</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E18:102_5_1_2_1_1_1_1_2_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="jordan2c.html#T35" target="_self">Th35</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">Q</font> = <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> <font color="Maroon">n</font></span>)</span> :  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">r</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">a</font> ) </span>}</span>  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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><b>reconsider </b><font color="Maroon">Q</font> = <font color="Maroon">Q</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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> ;<br/><a NAME="E5:102_5_1_2_1_1_1_1_2_1_2"/>
<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> <font color="Maroon">n</font></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#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> </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> <font color="Maroon">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_5_1_2_1_1_1_1_2_1_2_2"><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="E1:102_5_1_2_1_1_1_1_2_1_2_2"/>
<font color="Maroon">z</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> <font color="Maroon">n</font></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#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">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> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E3:102_5_1_2_1_1_1_1_2_1_2_2"/><i><font color="Green">E374</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> )
 ;<br/>
<b>thus </b><a NAME="E4:102_5_1_2_1_1_1_1_2_1_2_2"/>
<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> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T35" target="_self">Th35</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">P2</font> = <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> <font color="Maroon">n</font></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#NR4" target="_self">&gt;</a> <font color="Maroon">a</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> <font color="Maroon">n</font></span>)</span> ;<br/><a NAME="E7:102_5_1_2_1_1_1_1_2_1_2"/>
<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> <font color="Maroon">n</font></span>)</span> :  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">r</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">a</font> ) </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> <font color="Maroon">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_5_1_2_1_1_1_1_2_1_2_3"><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="E1:102_5_1_2_1_1_1_1_2_1_2_3"/>
<font color="Maroon">z</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> <font color="Maroon">n</font></span>)</span> :  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">r</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">a</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">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> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E3:102_5_1_2_1_1_1_1_2_1_2_3"/><i><font color="Green">E375</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp;  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">r</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">a</font> ) )
 ;<br/>
<b>thus </b><a NAME="E4:102_5_1_2_1_1_1_1_2_1_2_3"/>
<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> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T35" target="_self">Th35</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">P3</font> = <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> <font color="Maroon">n</font></span>)</span> :  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">r</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">a</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> <font color="Maroon">n</font></span>)</span> ;<br/><a NAME="E9:102_5_1_2_1_1_1_1_2_1_2"/>
<font color="Maroon">P2</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/><b>then reconsider </b><font color="Maroon">W2</font> = <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> <font color="Maroon">n</font></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#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> ;<br/><b>reconsider </b><font color="Maroon">W3</font> = <font color="Maroon">P3</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/><a NAME="E12:102_5_1_2_1_1_1_1_2_1_2"/>
<font color="Maroon">W3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P3</font>
 ;<br/><a NAME="E13:102_5_1_2_1_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E376</font></i>: 
<font color="Maroon">P3</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="jordan2c.html#T24" target="_self">Th24</a>, <a class="txt" href="jordan2c.html#E12:102"><i><font color="Green">E63</font></i></a></i>;<br/><a NAME="E14:102_5_1_2_1_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E377</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P3</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="E15:102_5_1_2_1_1_1_1_2_1_2"/><i><font color="Green">E378</font></i>: 
not <font color="Maroon">W3</font> is <a href="tbsp_1.html#V6">bounded</a>
 <b>by </b><i><a class="ref" href="jordan2c.html#T24" target="_self">Th24</a>, <a class="txt" href="jordan2c.html#E12:102"><i><font color="Green">E63</font></i></a></i>;<br/><div><i><font color="Green">E379</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="102_5_1_2_1_1_1_1_2_1_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:102_5_1_2_1_1_1_1_2_1_2_4"/><i><font color="Green">E380</font></i>: 
not <font color="Maroon">W2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">W2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>;<br/><a NAME="E3:102_5_1_2_1_1_1_1_2_1_2_4"/><i><font color="Green">E381</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="jordan2c.html#T39" target="_self">Th39</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:102_5_1_2_1_1_1_1_2_1_2_4"/><b>then </b><i><font color="Green">E382</font></i>: 
 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> <font color="Maroon">n</font></span>)</span> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</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#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> )
 ;<br/><a NAME="E5:102_5_1_2_1_1_1_1_2_1_2_4"/>
 ex <font color="Olive">q2</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> <font color="Maroon">n</font></span>)</span> st <br/>( <font color="Olive">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E6:102_5_1_2_1_1_1_1_2_1_2_4"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E17:102_5_1_2_1_1_1_1_2_1_2"/>
<font color="Maroon">W3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_5_1_2_1_1_1_1_2_1_2_5"><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="E1:102_5_1_2_1_1_1_1_2_1_2_5"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W3</font>
 ;<br/>

<b>then consider </b><font color="Maroon">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> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E3:102_5_1_2_1_1_1_1_2_1_2_5"/><i><font color="Green">E383</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp;  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">r</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">a</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:102_5_1_2_1_1_1_1_2_1_2_5"/><i><font color="Green">E384</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="jordan2c.html#T39" target="_self">Th39</a></i>;<br/>
<a NAME="E6:102_5_1_2_1_1_1_1_2_1_2_5"/><i><font color="Green">E385</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> 0
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E7:102_5_1_2_1_1_1_1_2_1_2_5"/><i><font color="Green">E386</font></i>: 
 <a href="binop_2.html#K7">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<b>reconsider </b><font color="Maroon">xr</font> = <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="binarith.html#K11">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="jordan2c.html#T24" target="_self">Th24</a></i>;<br/>
<i><font color="Green">E387</font></i>: <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> = 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">xr</font></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i>, <a class="ref" href="jgraph_1.html#D5">JGRAPH_1:def 5</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">xr</font></span>)</span></span>)</span>

;<br/>
<a NAME="E10:102_5_1_2_1_1_1_1_2_1_2_5"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">xr</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E11:102_5_1_2_1_1_1_1_2_1_2_5"/><b>then </b>
<font color="Maroon">xr</font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">xr</font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i><a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E12:102_5_1_2_1_1_1_1_2_1_2_5"/><b>then </b><i><font color="Green">E388</font></i>: 
 <a href="jordan2b.html#K1">Proj</a> <font color="Maroon">q</font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">xr</font> <a href="goboard1.html#K1">.</a> 1
 
<b>by </b><i>, <a class="ref" href="jordan2b.html#D1">JORDAN2B:def 1</a></i>;<br/>
<a NAME="E13:102_5_1_2_1_1_1_1_2_1_2_5"/><i><font color="Green">E389</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">xr</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="jordan2c.html#T24" target="_self">Th24</a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E14:102_5_1_2_1_1_1_1_2_1_2_5"/>
<span class="p1">(<span class="default"><a href="euclid.html#K11">sqr</a> <font color="Maroon">xr</font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">q</font>,1</span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T79">RVSUM_1:79</a></i>;<br/>
<a NAME="E15:102_5_1_2_1_1_1_1_2_1_2_5"/><b>then </b><i><font color="Green">E390</font></i>: 
 <a href="euclid.html#K11">sqr</a> <font color="Maroon">xr</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">q</font>,1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">q</font>,1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="jordan2b.html#K1">Proj</a> <font color="Maroon">q</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T158">COMPLEX1:158</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/>
<a NAME="E17:102_5_1_2_1_1_1_1_2_1_2_5"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T40" target="_self">Th40</a>, , <a class="ref" href="finsop_1.html#T12">FINSOP_1:12</a></i>;<br/>
<a NAME="E18:102_5_1_2_1_1_1_1_2_1_2_5"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<b>hence </b><a NAME="E19:102_5_1_2_1_1_1_1_2_1_2_5"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T39" target="_self">Th39</a></i>;<br/>


</div><b>end;</b></div><a NAME="E18:102_5_1_2_1_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E391</font></i>: 
<font color="Maroon">W3</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E19:102_5_1_2_1_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">W3</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="jordan2c.html#T38" target="_self">Th38</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><a NAME="E20:102_5_1_2_1_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E392</font></i>: 
<font color="Maroon">W3</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E21:102_5_1_2_1_1_1_1_2_1_2"/>
<font color="Maroon">W3</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="jordan2c.html#T38" target="_self">Th38</a>, <a class="ref" href="jordan2c.html#T39" target="_self">Th39</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><a NAME="E22:102_5_1_2_1_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">W3</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T32">SUBSET_1:32</a></i>;<br/><a NAME="E23:102_5_1_2_1_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E393</font></i>: 
<font color="Maroon">W3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><a NAME="E24:102_5_1_2_1_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E394</font></i>: 
<font color="Maroon">P3</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P3</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E25:102_5_1_2_1_1_1_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Q</font>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#T28">PRE_TOPC:28</a></i>;<br/><a NAME="E26:102_5_1_2_1_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E395</font></i>: 
<font color="Maroon">Q</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="jordan2c.html#T36" target="_self">Th36</a>, <a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a></i>;<br/><a NAME="E27:102_5_1_2_1_1_1_1_2_1_2"/>
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="connsp_3.html#K4">Down</a> <font color="Maroon">P3</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i/>;<br/><a NAME="E28:102_5_1_2_1_1_1_1_2_1_2"/><b>then </b>
 <a href="connsp_3.html#K5">Up</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K1">Component_of</a> <font color="Maroon">Q</font></span>)</span> <a href="jordan2c.html#R2" target="_self">is_outside_component_of</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T35" target="_self">Th35</a>, <a class="ref" href="jordan2c.html#T37" target="_self">Th37</a>, , <a class="txt" href="jordan2c.html#E18:102"><i><font color="Green">E66</font></i></a></i>;<br/><a NAME="E29:102_5_1_2_1_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E396</font></i>: 
 <a href="connsp_3.html#K1">Component_of</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><a NAME="E30:102_5_1_2_1_1_1_1_2_1_2"/>
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="connsp_3.html#K1">Component_of</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T40" target="_self">Th40</a>, <a class="ref" href="connsp_3.html#T1">CONNSP_3:1</a></i>;<br/><a NAME="E31:102_5_1_2_1_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E397</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="connsp_3.html#K1">Component_of</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T33" target="_self">Th33</a></i>;<br/><a NAME="E32:102_5_1_2_1_1_1_1_2_1_2"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/><a NAME="E33:102_5_1_2_1_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E34:102_5_1_2_1_1_1_1_2_1_2"/><b>then </b>
 <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E35:102_5_1_2_1_1_1_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E10:102_5_1_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_5_1_2_1_1_1_2"><b>case </b></a><a NAME="E1:102_5_1_2_1_1_1_2"/><i><font color="Green">E398</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q1</font> where <font color="Olive">q1</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> <font color="Maroon">n</font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:102_5_1_2_1_1_1_2"/>
 <a href="connsp_3.html#K4">Down</a> <font color="Maroon">P</font>,<span class="p1">(<span class="default"><font color="Maroon">A</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">P</font>,<span class="p2">(<span class="default"><font color="Maroon">A</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/><b>hence </b><a NAME="E3:102_5_1_2_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T29" target="_self">Th29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E18:102_5_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E6:102_5_1_2"/><b>then </b>
<font color="Maroon">P1</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> <font color="Maroon">n</font></span>)</span> : <font color="Olive">B</font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">A</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#D10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E7:102_5_1_2"/>
 ex <font color="Olive">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> <font color="Maroon">n</font></span>)</span> st <br/>( <font color="Olive">B</font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">A</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R1">=</a>  <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="jordan2c.html#T22" target="_self">Th22</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E26:102"/>
 <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">A</font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">A</font>
 ;<br/>


</div>
