<?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>assume </b><a NAME="E1:122"/>
<font color="Maroon">A</font> is <a href="jordan2c.html#V1" target="_self">Bounded</a>
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:122"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>  st <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:122_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_1"><b>suppose </b></a><a NAME="E1:122_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/><div class="add"><b>set </b><font color="Maroon">a</font> = <font color="Maroon">r</font>;<br/><b>reconsider </b><font color="Maroon">P</font> = <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <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">r</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> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/><a NAME="E3:122_1_1"/><i><font color="Green">E57</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="122_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="E1:122_1_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E2:122_1_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> &amp; not <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#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> </span>}</span>  )
 
<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">q0</font> = <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="jordan2c.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E4:122_1_1_1"/>
<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#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E5:122_1_1_1"/><b>then </b>
not <font color="Maroon">q0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E6:122_1_1_1"/><b>then </b>
<font color="Maroon">q0</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="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E7:122_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> 
 ;<br/>


</div><b>end;</b></div><a NAME="E4:122_1_1"/><b>then </b><i><font color="Green">E60</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_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="E1:122_1_1_2_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="122_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:122_1_1_2_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 ;<br/><div class="add"><a NAME="E2:122_1_1_2_1_1"/><b>then </b>
<font color="Maroon">P</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:122_1_1"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E3:122_1_1_2_1_1"/><b>then </b><i><font color="Green">E61</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="jordan2c.html#T28" target="_self">Th28</a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:122_1_1_2_1_1_1"/>
not  <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">A</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="E3:122_1_1_2_1_1_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">A</font> &amp; not <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">r</font> )
 <b>by </b><i/>;<br/><a NAME="E4:122_1_1_2_1_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">B3</font> where <font color="Olive">B3</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">B3</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#T30" target="_self">Th30</a></i>;<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="E6:122_1_1_2_1_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">q</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">B3</font> where <font color="Olive">B3</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">B3</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">B3</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="E8:122_1_1_2_1_1_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B3</font> &amp; <font color="Maroon">B3</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#T31" target="_self">Th31</a></i>;<br/><a NAME="E9:122_1_1_2_1_1_1"/>
<font color="Maroon">B3</font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">B4</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="E11:122_1_1_2_1_1_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">B4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B3</font> &amp; <font color="Maroon">B4</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> )
 <b>by </b><i><a class="ref" href="connsp_1.html#D6">CONNSP_1:def 6</a></i>;<br/><a NAME="E12:122_1_1_2_1_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">q</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>
 ;<br/><a NAME="E13:122_1_1_2_1_1_1"/>
for <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 <font color="Olive">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> holds <br/><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="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T30" target="_self">Th30</a></i>;<br/><a NAME="E14:122_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> &amp; not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q2</font> where <font color="Olive">q2</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">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> </span>}</span>  )
 <b>by </b><i><a class="ref" href="jordan2c.html#T35" target="_self">Th35</a>, <a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/><a NAME="E15:122_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E16:122_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B4</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <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#T31" target="_self">Th31</a>, , <a class="ref" href="jordan2c.html#T33" target="_self">Th33</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E17:122_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">P</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B4</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E18:122_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E130</font></i>: 
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B4</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T28" target="_self">Th28</a>, <a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, <a class="ref" href="jordan2c.html#T33" target="_self">Th33</a>, <a class="ref" href="connsp_1.html#T38">CONNSP_1:38</a></i>;<br/><a NAME="E19:122_1_1_2_1_1_1"/>
<font color="Maroon">B3</font> is <a href="jordan2c.html#V1" target="_self">Bounded</a>
 <b>by </b><i>, </i>;<br/><a NAME="E20:122_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">P</font> is <a href="jordan2c.html#V1" target="_self">Bounded</a>
 <b>by </b><i><a class="ref" href="jordan2c.html#T33" target="_self">Th33</a>, <a class="ref" href="jordan2c.html#T36" target="_self">Th36</a>, </i>;<br/><b>hence </b><a NAME="E21:122_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan2c.html#E3:122_1_1"><i><font color="Green">E57</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:122_1_1_2_1_1"/>
 <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">A</font> is <a href="jordan2c.html#V1" target="_self">Bounded</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:122_1_1_2_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2
 ;<br/><div class="add"><a NAME="E2:122_1_1_2_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="E3:122_1_1_2_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="E4:122_1_1_2_1_2"/><b>then </b><i><font color="Green">E131</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="E5:122_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> : for <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Olive">r2</font></span><a href="jordan2b.html#K2">]|</a></span> holds <br/><font color="Olive">r2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">r</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="122_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="E1:122_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> : for <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Olive">r2</font></span><a href="jordan2b.html#K2">]|</a></span> holds <br/><font color="Olive">r2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">r</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="E3:122_1_1_2_1_2_1"/><i><font color="Green">E152</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; ( for <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Olive">r2</font></span><a href="jordan2b.html#K2">]|</a></span> holds <br/><font color="Olive">r2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">r</font> ) )
 ;<br/>
<b>thus </b><a NAME="E4:122_1_1_2_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <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">P1</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> : for <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Olive">r2</font></span><a href="jordan2b.html#K2">]|</a></span> holds <br/><font color="Olive">r2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">r</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="E7:122_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> : for <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Olive">r2</font></span><a href="jordan2b.html#K2">]|</a></span> holds <br/><font color="Olive">r2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</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="122_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="E1:122_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> : for <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Olive">r2</font></span><a href="jordan2b.html#K2">]|</a></span> holds <br/><font color="Olive">r2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</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="E3:122_1_1_2_1_2_2"/><i><font color="Green">E154</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; ( for <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Olive">r2</font></span><a href="jordan2b.html#K2">]|</a></span> holds <br/><font color="Olive">r2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font> ) )
 ;<br/>
<b>thus </b><a NAME="E4:122_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#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> : for <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Olive">r2</font></span><a href="jordan2b.html#K2">]|</a></span> holds <br/><font color="Olive">r2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</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="E9:122_1_1_2_1_2"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_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="E1:122_1_1_2_1_2_3"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E2:122_1_1_2_1_2_3"/><b>then </b><i><font color="Green">E159</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> &amp; not <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#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> </span>}</span>  )
 
<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">q0</font> = <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="jordan2c.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E4:122_1_1_2_1_2_3"/><i><font color="Green">E160</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="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">r3</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:122_1_1_2_1_2_3"/><i><font color="Green">E161</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">r3</font></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, <a class="ref" href="jordan2b.html#T24">JORDAN2B:24</a></i>;<br/>
<a NAME="E7:122_1_1_2_1_2_3"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">q0</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Maroon">r3</font></span><a href="jordan2b.html#K2">]|</a></span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T35" target="_self">Th35</a>, <a class="ref" href="jordan2b.html#D2">JORDAN2B:def 2</a></i>;<br/>
<a NAME="E8:122_1_1_2_1_2_3"/><b>then </b><i><font color="Green">E163</font></i>: 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">r3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, <a class="ref" href="jordan2c.html#T33" target="_self">Th33</a>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_1_2_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:122_1_1_2_1_2_3_1"/>
(  <a href="binop_2.html#K7">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r3</font> or <font color="Maroon">r3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font> )
 </span><b>by </b><i><a class="ref" href="jordan2c.html#T37" target="_self">Th37</a>, <a class="ref" href="seq_2.html#T9">SEQ_2:9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_1_2_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:122_1_1_2_1_2_3_1_1"/>
 <a href="binop_2.html#K7">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r3</font>
 ;<br/><div class="add"><a NAME="E2:122_1_1_2_1_2_3_1_1"/><b>then </b>
for <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">q0</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Olive">r2</font></span><a href="jordan2b.html#K2">]|</a></span> holds <br/><font color="Olive">r2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T36" target="_self">Th36</a>, <a class="ref" href="jordan2b.html#T29">JORDAN2B:29</a></i>;<br/><a NAME="E3:122_1_1_2_1_2_3_1_1"/><b>then </b>
<font color="Maroon">q0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 ;<br/><b>hence </b><a NAME="E4:122_1_1_2_1_2_3_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_1_2_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:122_1_1_2_1_2_3_1_2"/>
<font color="Maroon">r3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><a NAME="E2:122_1_1_2_1_2_3_1_2"/><b>then </b>
for <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">q0</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Olive">r2</font></span><a href="jordan2b.html#K2">]|</a></span> holds <br/><font color="Olive">r2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T36" target="_self">Th36</a>, <a class="ref" href="jordan2b.html#T29">JORDAN2B:29</a></i>;<br/><a NAME="E3:122_1_1_2_1_2_3_1_2"/><b>then </b>
<font color="Maroon">q0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font>
 ;<br/><b>hence </b><a NAME="E4:122_1_1_2_1_2_3_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E10:122_1_1_2_1_2"/>
<font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_1_2_1_2_4"><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="E1:122_1_1_2_1_2_4"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P2</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_1_2_1_2_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:122_1_1_2_1_2_4_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> or <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font> )
 </span><b>by </b><i><a class="ref" href="jordan2c.html#T31" target="_self">Th31</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_1_2_1_2_4_1_1"><b>suppose </b></a><a NAME="E1:122_1_1_2_1_2_4_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 ;<br/><div class="add"><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="E3:122_1_1_2_1_2_4_1_1"/><i><font color="Green">E165</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; ( for <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Olive">r2</font></span><a href="jordan2b.html#K2">]|</a></span> holds <br/><font color="Olive">r2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">r</font> ) )
 ;<br/><a NAME="E4:122_1_1_2_1_2_4_1_1"/><i><font color="Green">E166</font></i>: 
<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/>;<br/><a NAME="E5:122_1_1_2_1_2_4_1_1"/>
for <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 <font color="Olive">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> holds <br/><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="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_1_2_1_2_4_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:122_1_1_2_1_2_4_1_1_1"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/>

<b>consider </b><font color="Maroon">r3</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:122_1_1_2_1_2_4_1_1_1"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">r3</font></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, <a class="ref" href="jordan2b.html#T24">JORDAN2B:24</a></i>;<br/>
<a NAME="E4:122_1_1_2_1_2_4_1_1_1"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Maroon">r3</font></span><a href="jordan2b.html#K2">]|</a></span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T36" target="_self">Th36</a>, <a class="ref" href="jordan2b.html#D2">JORDAN2B:def 2</a></i>;<br/>
<a NAME="E5:122_1_1_2_1_2_4_1_1_1"/><b>then </b><i><font color="Green">E170</font></i>: 
<font color="Maroon">r3</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="jordan2c.html#T35" target="_self">Th35</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_1_2_1_2_4_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_1_2_1_2_4_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="E1:122_1_1_2_1_2_4_1_1_1_1_1"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or <font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_1_2_1_2_4_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:122_1_1_2_1_2_4_1_1_1_1_1_1"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:122_1_1_2_1_2_4_1_1_1_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">r3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><b>hence </b><a NAME="E3:122_1_1_2_1_2_4_1_1_1_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">r3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T39" target="_self">Th39</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_1_2_1_2_4_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:122_1_1_2_1_2_4_1_1_1_1_1_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:122_1_1_2_1_2_4_1_1_1_1_1_2"/><b>then </b>
 <a href="binop_2.html#K7">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&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="E3:122_1_1_2_1_2_4_1_1_1_1_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">r3</font> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">r3</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T38" target="_self">Th38</a>, </i>;<br/><b>hence </b><a NAME="E4:122_1_1_2_1_2_4_1_1_1_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">r3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</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="real_2.html#T109">REAL_2:109</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:122_1_1_2_1_2_4_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="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, <a class="ref" href="jordan2c.html#T37" target="_self">Th37</a>, </i>;<br/>


</div><b>end;</b></div><a NAME="E6:122_1_1_2_1_2_4_1_1"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> &amp; not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q2</font> where <font color="Olive">q2</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">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> </span>}</span>  )
 <b>by </b><i><a class="ref" href="jordan2c.html#T33" target="_self">Th33</a>, <a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/><b>hence </b><a NAME="E7:122_1_1_2_1_2_4_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_1_2_1_2_4_1_2"><b>suppose </b></a><a NAME="E1:122_1_1_2_1_2_4_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font>
 ;<br/><div class="add"><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="E3:122_1_1_2_1_2_4_1_2"/><i><font color="Green">E172</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; ( for <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Olive">r2</font></span><a href="jordan2b.html#K2">]|</a></span> holds <br/><font color="Olive">r2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font> ) )
 ;<br/><a NAME="E4:122_1_1_2_1_2_4_1_2"/><i><font color="Green">E205</font></i>: 
<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/>;<br/><a NAME="E5:122_1_1_2_1_2_4_1_2"/>
for <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 <font color="Olive">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> holds <br/><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="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_1_2_1_2_4_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:122_1_1_2_1_2_4_1_2_1"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/>

<b>consider </b><font color="Maroon">r3</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:122_1_1_2_1_2_4_1_2_1"/><i><font color="Green">E207</font></i>: 
<font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">r3</font></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, <a class="ref" href="jordan2b.html#T24">JORDAN2B:24</a></i>;<br/>
<a NAME="E4:122_1_1_2_1_2_4_1_2_1"/><i><font color="Green">E208</font></i>: 
<font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Maroon">r3</font></span><a href="jordan2b.html#K2">]|</a></span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T36" target="_self">Th36</a>, <a class="ref" href="jordan2b.html#D2">JORDAN2B:def 2</a></i>;<br/>
<a NAME="E5:122_1_1_2_1_2_4_1_2_1"/><b>then </b><i><font color="Green">E209</font></i>: 
<font color="Maroon">r3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="jordan2c.html#T35" target="_self">Th35</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_1_2_1_2_4_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_1_2_1_2_4_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="E1:122_1_1_2_1_2_4_1_2_1_1_1"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or <font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_1_2_1_2_4_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:122_1_1_2_1_2_4_1_2_1_1_1_1"/><i><font color="Green">E210</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:122_1_1_2_1_2_4_1_2_1_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">r3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><b>hence </b><a NAME="E3:122_1_1_2_1_2_4_1_2_1_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">r3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T39" target="_self">Th39</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_1_2_1_2_4_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:122_1_1_2_1_2_4_1_2_1_1_1_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:122_1_1_2_1_2_4_1_2_1_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">r3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</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="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:122_1_1_2_1_2_4_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="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, <a class="ref" href="jordan2c.html#T37" target="_self">Th37</a>, </i>;<br/>


</div><b>end;</b></div><a NAME="E6:122_1_1_2_1_2_4_1_2"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> &amp; not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q2</font> where <font color="Olive">q2</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">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> </span>}</span>  )
 <b>by </b><i><a class="ref" href="jordan2c.html#T33" target="_self">Th33</a>, <a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/><b>hence </b><a NAME="E7:122_1_1_2_1_2_4_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E11:122_1_1_2_1_2"/><b>then </b><i><font color="Green">E211</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P2</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T30" target="_self">Th30</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E12:122_1_1_2_1_2"/><b>then </b>
<font color="Maroon">P1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E13:122_1_1_2_1_2"/><b>then </b>
<font color="Maroon">P1</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="jordan2c.html#T27" target="_self">Th27</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E14:122_1_1_2_1_2"/><b>then </b><i><font color="Green">E212</font></i>: 
 <a href="connsp_3.html#K4">Down</a> <font color="Maroon">P1</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">P1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E15:122_1_1_2_1_2"/>
<font color="Maroon">P2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T31" target="_self">Th31</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E16:122_1_1_2_1_2"/><b>then </b>
<font color="Maroon">P2</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="jordan2c.html#T27" target="_self">Th27</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E17:122_1_1_2_1_2"/><b>then </b><i><font color="Green">E213</font></i>: 
 <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> <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="E18:122_1_1_2_1_2"/>
for <font color="Olive">w1</font>, <font color="Olive">w2</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 <font color="Olive">w1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> &amp; <font color="Olive">w2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> holds <br/> <a href="topreal1.html#K3">LSeg</a> <font color="Olive">w1</font>,<font color="Olive">w2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_1_2_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">w2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:122_1_1_2_1_2_5"/><i><font color="Green">E214</font></i>: 
( <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> &amp; <font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> )
 ;<br/>

<b>then 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="E3:122_1_1_2_1_2_5"/><i><font color="Green">E215</font></i>: 
( <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> &amp; ( for <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Olive">r2</font></span><a href="jordan2b.html#K2">]|</a></span> holds <br/><font color="Olive">r2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">r</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">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><b> such that </b><br/><a NAME="E5:122_1_1_2_1_2_5"/><i><font color="Green">E216</font></i>: 
( <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w2</font> &amp; ( for <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Olive">r2</font></span><a href="jordan2b.html#K2">]|</a></span> holds <br/><font color="Olive">r2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">r</font> ) )
 <b>by </b><i><a class="ref" href="jordan2c.html#T35" target="_self">Th35</a></i>;<br/>
<b>consider </b><font color="Maroon">r3</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:122_1_1_2_1_2_5"/><i><font color="Green">E217</font></i>: 
<font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">r3</font></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, <a class="ref" href="jordan2b.html#T24">JORDAN2B:24</a></i>;<br/>
<a NAME="E8:122_1_1_2_1_2_5"/><i><font color="Green">E218</font></i>: 
<font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Maroon">r3</font></span><a href="jordan2b.html#K2">]|</a></span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T38" target="_self">Th38</a>, <a class="ref" href="jordan2b.html#D2">JORDAN2B:def 2</a></i>;<br/>
<a NAME="E9:122_1_1_2_1_2_5"/><b>then </b><i><font color="Green">E219</font></i>: 
<font color="Maroon">r3</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T36" target="_self">Th36</a></i>;<br/>
<b>consider </b><font color="Maroon">r4</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E11:122_1_1_2_1_2_5"/><i><font color="Green">E220</font></i>: 
<font color="Maroon">w2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">r4</font></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, <a class="ref" href="jordan2b.html#T24">JORDAN2B:24</a></i>;<br/>
<a NAME="E12:122_1_1_2_1_2_5"/><i><font color="Green">E221</font></i>: 
<font color="Maroon">w2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Maroon">r4</font></span><a href="jordan2b.html#K2">]|</a></span>
 
<b>by </b><i>, <a class="ref" href="jordan2b.html#D2">JORDAN2B:def 2</a></i>;<br/>
<a NAME="E13:122_1_1_2_1_2_5"/><b>then </b><i><font color="Green">E222</font></i>: 
<font color="Maroon">r4</font> <a href="xxreal_0.html#R1">&lt;=</a>  <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></i>;<br/>
<b>thus </b><a NAME="E14:122_1_1_2_1_2_5"/>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w1</font>,<font color="Maroon">w2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_1_2_1_2_5_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="E1:122_1_1_2_1_2_5_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w1</font>,<font color="Maroon">w2</font>
 ;<br/>

<a NAME="E2:122_1_1_2_1_2_5_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Olive">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w2</font></span>)</span></span>)</span> where <font color="Olive">r2</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r2</font> &amp; <font color="Olive">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="topreal1.html#D3">TOPREAL1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:122_1_1_2_1_2_5_1"/><i><font color="Green">E223</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w2</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>
<a NAME="E5:122_1_1_2_1_2_5_1"/><i><font color="Green">E224</font></i>: 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r2</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">r3</font></span>)</span></span><a href="jordan2b.html#K2">]|</a></span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, <a class="ref" href="jordan2c.html#T39" target="_self">Th39</a>, <a class="ref" href="jordan2b.html#T26">JORDAN2B:26</a></i>;<br/>
<a NAME="E6:122_1_1_2_1_2_5_1"/>
<font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r4</font></span>)</span></span><a href="jordan2b.html#K2">]|</a></span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, , <a class="ref" href="jordan2b.html#T26">JORDAN2B:26</a></i>;<br/>
<a NAME="E7:122_1_1_2_1_2_5_1"/><b>then </b><i><font color="Green">E225</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r2</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">r3</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r4</font></span>)</span></span>)</span></span><a href="jordan2b.html#K2">]|</a></span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, , , <a class="ref" href="jordan2b.html#T27">JORDAN2B:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q4</font> = <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E9:122_1_1_2_1_2_5_1"/>
for <font color="Olive">r5</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">q4</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Olive">r5</font></span><a href="jordan2b.html#K2">]|</a></span> holds <br/><font color="Olive">r5</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_1_2_1_2_5_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r5</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:122_1_1_2_1_2_5_1_1"/>
<font color="Maroon">q4</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Maroon">r5</font></span><a href="jordan2b.html#K2">]|</a></span>
 ;<br/>

<a NAME="E2:122_1_1_2_1_2_5_1_1"/><b>then </b><i><font color="Green">E232</font></i>: 
<font color="Maroon">r5</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r2</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">r3</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r4</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan2b.html#T29">JORDAN2B:29</a></i>;<br/>
<a NAME="E3:122_1_1_2_1_2_5_1_1"/>
1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E4:122_1_1_2_1_2_5_1_1"/><b>then </b><i><font color="Green">E322</font></i>: 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r2</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">r3</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r2</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E5:122_1_1_2_1_2_5_1_1"/><i><font color="Green">E323</font></i>: 
<font color="Maroon">r2</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r4</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T40" target="_self">Th40</a>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E6:122_1_1_2_1_2_5_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r2</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">r</font>
 
;<br/>
<b>hence </b><a NAME="E7:122_1_1_2_1_2_5_1_1"/>
<font color="Maroon">r5</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2"><i><font color="Green">E131</font></i></a>, <a class="ref" href="jordan2c.html#T41" target="_self">Th41</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:122_1_1_2_1_2_5_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 ;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div><a NAME="E19:122_1_1_2_1_2"/><b>then </b>
<font color="Maroon">P1</font> is <a href="jordan1.html#V1">convex</a>
 <b>by </b><i><a class="ref" href="jordan1.html#D1">JORDAN1:def 1</a></i>;<br/><a NAME="E20:122_1_1_2_1_2"/><b>then </b><i><font color="Green">E324</font></i>: 
<font color="Maroon">P1</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="jordan1.html#T9">JORDAN1:9</a></i>;<br/><a NAME="E21:122_1_1_2_1_2"/>
for <font color="Olive">w1</font>, <font color="Olive">w2</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 <font color="Olive">w1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font> &amp; <font color="Olive">w2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font> holds <br/> <a href="topreal1.html#K3">LSeg</a> <font color="Olive">w1</font>,<font color="Olive">w2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_1_2_1_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">w2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:122_1_1_2_1_2_6"/><i><font color="Green">E325</font></i>: 
( <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font> &amp; <font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font> )
 ;<br/>

<b>then 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="E3:122_1_1_2_1_2_6"/><i><font color="Green">E326</font></i>: 
( <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> &amp; ( for <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Olive">r2</font></span><a href="jordan2b.html#K2">]|</a></span> holds <br/><font color="Olive">r2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">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><b> such that </b><br/><a NAME="E5:122_1_1_2_1_2_6"/><i><font color="Green">E327</font></i>: 
( <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w2</font> &amp; ( for <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Olive">r2</font></span><a href="jordan2b.html#K2">]|</a></span> holds <br/><font color="Olive">r2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font> ) )
 <b>by </b><i><a class="ref" href="jordan2c.html#T36" target="_self">Th36</a></i>;<br/>
<b>consider </b><font color="Maroon">r3</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:122_1_1_2_1_2_6"/><i><font color="Green">E328</font></i>: 
<font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">r3</font></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, <a class="ref" href="jordan2b.html#T24">JORDAN2B:24</a></i>;<br/>
<a NAME="E8:122_1_1_2_1_2_6"/><i><font color="Green">E329</font></i>: 
<font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Maroon">r3</font></span><a href="jordan2b.html#K2">]|</a></span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T39" target="_self">Th39</a>, <a class="ref" href="jordan2b.html#D2">JORDAN2B:def 2</a></i>;<br/>
<a NAME="E9:122_1_1_2_1_2_6"/><b>then </b><i><font color="Green">E330</font></i>: 
<font color="Maroon">r3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T37" target="_self">Th37</a></i>;<br/>
<b>consider </b><font color="Maroon">r4</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E11:122_1_1_2_1_2_6"/><i><font color="Green">E331</font></i>: 
<font color="Maroon">w2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">r4</font></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, <a class="ref" href="jordan2b.html#T24">JORDAN2B:24</a></i>;<br/>
<a NAME="E12:122_1_1_2_1_2_6"/><i><font color="Green">E332</font></i>: 
<font color="Maroon">w2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Maroon">r4</font></span><a href="jordan2b.html#K2">]|</a></span>
 
<b>by </b><i>, <a class="ref" href="jordan2b.html#D2">JORDAN2B:def 2</a></i>;<br/>
<a NAME="E13:122_1_1_2_1_2_6"/><b>then </b><i><font color="Green">E333</font></i>: 
<font color="Maroon">r4</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T38" target="_self">Th38</a></i>;<br/>
<b>thus </b><a NAME="E14:122_1_1_2_1_2_6"/>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w1</font>,<font color="Maroon">w2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_1_2_1_2_6_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="E1:122_1_1_2_1_2_6_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w1</font>,<font color="Maroon">w2</font>
 ;<br/>

<a NAME="E2:122_1_1_2_1_2_6_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Olive">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w2</font></span>)</span></span>)</span> where <font color="Olive">r2</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r2</font> &amp; <font color="Olive">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="topreal1.html#D3">TOPREAL1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:122_1_1_2_1_2_6_1"/><i><font color="Green">E334</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w2</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>
<a NAME="E5:122_1_1_2_1_2_6_1"/><i><font color="Green">E335</font></i>: 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r2</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">r3</font></span>)</span></span><a href="jordan2b.html#K2">]|</a></span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, , <a class="ref" href="jordan2b.html#T26">JORDAN2B:26</a></i>;<br/>
<a NAME="E6:122_1_1_2_1_2_6_1"/>
<font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r4</font></span>)</span></span><a href="jordan2b.html#K2">]|</a></span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, <a class="ref" href="jordan2c.html#T40" target="_self">Th40</a>, <a class="ref" href="jordan2b.html#T26">JORDAN2B:26</a></i>;<br/>
<a NAME="E7:122_1_1_2_1_2_6_1"/><b>then </b><i><font color="Green">E336</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r2</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">r3</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r4</font></span>)</span></span>)</span></span><a href="jordan2b.html#K2">]|</a></span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, , , <a class="ref" href="jordan2b.html#T27">JORDAN2B:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q4</font> = <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E9:122_1_1_2_1_2_6_1"/>
for <font color="Olive">r5</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">q4</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Olive">r5</font></span><a href="jordan2b.html#K2">]|</a></span> holds <br/><font color="Olive">r5</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_1_2_1_2_6_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r5</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:122_1_1_2_1_2_6_1_1"/>
<font color="Maroon">q4</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Maroon">r5</font></span><a href="jordan2b.html#K2">]|</a></span>
 ;<br/>

<a NAME="E2:122_1_1_2_1_2_6_1_1"/><b>then </b><i><font color="Green">E337</font></i>: 
<font color="Maroon">r5</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r2</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">r3</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r4</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan2b.html#T29">JORDAN2B:29</a></i>;<br/>
<a NAME="E3:122_1_1_2_1_2_6_1_1"/>
1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E4:122_1_1_2_1_2_6_1_1"/><b>then </b><i><font color="Green">E338</font></i>: 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r2</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">r3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r2</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E5:122_1_1_2_1_2_6_1_1"/><i><font color="Green">E339</font></i>: 
<font color="Maroon">r2</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r4</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r2</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E6:122_1_1_2_1_2_6_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r2</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 
;<br/>
<b>hence </b><a NAME="E7:122_1_1_2_1_2_6_1_1"/>
<font color="Maroon">r5</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2"><i><font color="Green">E131</font></i></a>, <a class="ref" href="jordan2c.html#T41" target="_self">Th41</a>, <a class="ref" href="jordan2c.html#T42" target="_self">Th42</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:122_1_1_2_1_2_6_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font>
 ;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div><a NAME="E22:122_1_1_2_1_2"/><b>then </b>
<font color="Maroon">P2</font> is <a href="jordan1.html#V1">convex</a>
 <b>by </b><i><a class="ref" href="jordan1.html#D1">JORDAN1:def 1</a></i>;<br/><a NAME="E23:122_1_1_2_1_2"/><b>then </b><i><font color="Green">E340</font></i>: 
<font color="Maroon">P2</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="jordan1.html#T9">JORDAN1:9</a></i>;<br/><div><i><font color="Green">E341</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_1_2_1_2_7"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:122_1_1_2_1_2_7"/>
<font color="Maroon">P1</font> is <a href="jordan2c.html#V1" target="_self">Bounded</a>
 ;<br/><b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:122_1_1_2_1_2_7"/><i><font color="Green">E344</font></i>: 
for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>  st <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> holds <br/><span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">p3</font> = <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span><a href="jordan2b.html#K2">]|</a></span>;<br/><a NAME="E4:122_1_1_2_1_2_7"/>
for <font color="Olive">r5</font> being  <a href="real_1.html#NM1">Real</a>  st <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span><a href="jordan2b.html#K2">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Olive">r5</font></span><a href="jordan2b.html#K2">]|</a></span> holds <br/><font color="Olive">r5</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_1_2_1_2_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r5</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:122_1_1_2_1_2_7_1"/>
<span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span><a href="jordan2b.html#K2">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Maroon">r5</font></span><a href="jordan2b.html#K2">]|</a></span>
 ;<br/>

<a NAME="E2:122_1_1_2_1_2_7_1"/><b>then </b><i><font color="Green">E345</font></i>: 
<font color="Maroon">r5</font> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan2b.html#T29">JORDAN2B:29</a></i>;<br/>
<a NAME="E3:122_1_1_2_1_2_7_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/>
<a NAME="E4:122_1_1_2_1_2_7_1"/><b>then </b><i><font color="Green">E346</font></i>: 
 <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E5:122_1_1_2_1_2_7_1"/>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E6:122_1_1_2_1_2_7_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="real_2.html#T173">REAL_2:173</a></i>;<br/>
<a NAME="E7:122_1_1_2_1_2_7_1"/><b>then </b>
 <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<b>hence </b><a NAME="E8:122_1_1_2_1_2_7_1"/>
<font color="Maroon">r5</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="binop_2.html#K7">-</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="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:122_1_1_2_1_2_7"/><b>then </b>
<span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span><a href="jordan2b.html#K2">]|</a></span> <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/><a NAME="E6:122_1_1_2_1_2_7"/><b>then </b><i><font color="Green">E347</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2"><a href="jordan2b.html#K2">|[</a><span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span><a href="jordan2b.html#K2">]|</a></span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, <a class="ref" href="jordan2c.html#T38" target="_self">Th38</a></i>;<br/><i><font color="Green">E348</font></i>: <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2"><a href="jordan2b.html#K2">|[</a><span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span></span><a href="jordan2b.html#K2">]|</a></span></span><a href="toprns_1.html#K5">.|</a></span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
;<br/><a NAME="E8:122_1_1_2_1_2_7"/><i><font color="Green">E349</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><a NAME="E9:122_1_1_2_1_2_7"/><i><font color="Green">E350</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><a NAME="E10:122_1_1_2_1_2_7"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E11:122_1_1_2_1_2_7"/><b>then </b><i><font color="Green">E351</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E12:122_1_1_2_1_2_7"/><i><font color="Green">E352</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/><a NAME="E13:122_1_1_2_1_2_7"/>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="real_2.html#T173">REAL_2:173</a></i>;<br/><b>hence </b><a NAME="E14:122_1_1_2_1_2_7"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan2c.html#T39" target="_self">Th39</a>, , <a class="ref" href="jordan2c.html#T40" target="_self">Th40</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E353</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_1_2_1_2_8"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:122_1_1_2_1_2_8"/>
<font color="Maroon">P2</font> is <a href="jordan2c.html#V1" target="_self">Bounded</a>
 ;<br/><b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:122_1_1_2_1_2_8"/><i><font color="Green">E354</font></i>: 
for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>  st <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font> holds <br/><span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">p3</font> = <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span></span><a href="jordan2b.html#K2">]|</a></span>;<br/><a NAME="E4:122_1_1_2_1_2_8"/>
for <font color="Olive">r5</font> being  <a href="real_1.html#NM1">Real</a>  st <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span></span><a href="jordan2b.html#K2">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Olive">r5</font></span><a href="jordan2b.html#K2">]|</a></span> holds <br/><font color="Olive">r5</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_1_2_1_2_8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r5</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:122_1_1_2_1_2_8_1"/>
<span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span></span><a href="jordan2b.html#K2">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Maroon">r5</font></span><a href="jordan2b.html#K2">]|</a></span>
 ;<br/>

<a NAME="E2:122_1_1_2_1_2_8_1"/><b>then </b><i><font color="Green">E355</font></i>: 
<font color="Maroon">r5</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan2b.html#T29">JORDAN2B:29</a></i>;<br/>
<a NAME="E3:122_1_1_2_1_2_8_1"/><i><font color="Green">E356</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/>
<a NAME="E4:122_1_1_2_1_2_8_1"/>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E5:122_1_1_2_1_2_8_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="real_2.html#T173">REAL_2:173</a></i>;<br/>
<b>hence </b><a NAME="E6:122_1_1_2_1_2_8_1"/>
<font color="Maroon">r5</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:122_1_1_2_1_2_8"/><b>then </b>
<span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span></span><a href="jordan2b.html#K2">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a></i>;<br/><a NAME="E6:122_1_1_2_1_2_8"/><b>then </b><i><font color="Green">E357</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2"><a href="jordan2b.html#K2">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span></span><a href="jordan2b.html#K2">]|</a></span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a>, <a class="ref" href="jordan2c.html#T39" target="_self">Th39</a></i>;<br/><a NAME="E7:122_1_1_2_1_2_8"/><i><font color="Green">E358</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2"><a href="jordan2b.html#K2">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span></span><a href="jordan2b.html#K2">]|</a></span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E8:122_1_1_2_1_2_8"/><i><font color="Green">E359</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><a NAME="E9:122_1_1_2_1_2_8"/><i><font color="Green">E360</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><a NAME="E10:122_1_1_2_1_2_8"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E11:122_1_1_2_1_2_8"/><b>then </b><i><font color="Green">E361</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E12:122_1_1_2_1_2_8"/><i><font color="Green">E362</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/><a NAME="E13:122_1_1_2_1_2_8"/>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T40" target="_self">Th40</a>, <a class="ref" href="real_2.html#T173">REAL_2:173</a></i>;<br/><b>hence </b><a NAME="E14:122_1_1_2_1_2_8"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><a NAME="E26:122_1_1_2_1_2"/><i><font color="Green">E363</font></i>: 
 <a href="connsp_3.html#K4">Down</a> <font color="Maroon">P1</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="jordan2c.html#T35" target="_self">Th35</a>, </i>;<br/><a NAME="E27:122_1_1_2_1_2"/><i><font color="Green">E364</font></i>: 
 <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> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="jordan2c.html#T33" target="_self">Th33</a>, <a class="ref" href="jordan2c.html#T36" target="_self">Th36</a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_1_2_1_2_9"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:122_1_1_2_1_2_9"/>
not  <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">A</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="E3:122_1_1_2_1_2_9"/><i><font color="Green">E365</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">A</font> &amp; not <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">r</font> )
 <b>by </b><i/>;<br/><a NAME="E4:122_1_1_2_1_2_9"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">B3</font> where <font color="Olive">B3</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">B3</font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a> <font color="Maroon">A</font> </span>}</span> 
 <b>by </b><i/>;<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="E6:122_1_1_2_1_2_9"/><i><font color="Green">E366</font></i>: 
( <font color="Maroon">q</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">B3</font> where <font color="Olive">B3</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">B3</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">B3</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="E8:122_1_1_2_1_2_9"/><i><font color="Green">E367</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B3</font> &amp; <font color="Maroon">B3</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="E9:122_1_1_2_1_2_9"/>
<font color="Maroon">B3</font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="jordan2c.html#T40" target="_self">Th40</a>, </i>;<br/><b>then consider </b><font color="Maroon">B4</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="E11:122_1_1_2_1_2_9"/><i><font color="Green">E368</font></i>: 
( <font color="Maroon">B4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B3</font> &amp; <font color="Maroon">B4</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> )
 <b>by </b><i><a class="ref" href="connsp_1.html#D6">CONNSP_1:def 6</a></i>;<br/><a NAME="E12:122_1_1_2_1_2_9"/><i><font color="Green">E369</font></i>: 
<font color="Maroon">q</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>
 ;<br/><a NAME="E13:122_1_1_2_1_2_9"/>
for <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 <font color="Olive">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> holds <br/><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="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/><a NAME="E14:122_1_1_2_1_2_9"/><b>then </b>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> &amp; not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q2</font> where <font color="Olive">q2</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">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> </span>}</span>  )
 <b>by </b><i>, <a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/><a NAME="E15:122_1_1_2_1_2_9"/><b>then </b><i><font color="Green">E370</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_1_2_1_2_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:122_1_1_2_1_2_9_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> or <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font> )
 </span><b>by </b><i><a class="ref" href="jordan2c.html#T30" target="_self">Th30</a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_1_2_1_2_9_1_1"><b>suppose </b></a><a NAME="E1:122_1_1_2_1_2_9_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 ;<br/><div class="add"><a NAME="E2:122_1_1_2_1_2_9_1_1"/><b>then </b>
<font color="Maroon">P1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B4</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <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#T40" target="_self">Th40</a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:122_1_1_2_1_2_9_1_1"/><b>then </b>
<font color="Maroon">P1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B4</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E4:122_1_1_2_1_2_9_1_1"/><b>then </b><i><font color="Green">E371</font></i>: 
<font color="Maroon">P1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B4</font>
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T39" target="_self">Th39</a>, , <a class="ref" href="connsp_1.html#T38">CONNSP_1:38</a></i>;<br/><a NAME="E5:122_1_1_2_1_2_9_1_1"/>
<font color="Maroon">B3</font> is <a href="jordan2c.html#V1" target="_self">Bounded</a>
 <b>by </b><i><a class="ref" href="jordan2c.html#T40" target="_self">Th40</a>, </i>;<br/><b>hence </b><a NAME="E6:122_1_1_2_1_2_9_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan2c.html#T37" target="_self">Th37</a>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_1_2_1_2_9_1_2"><b>suppose </b></a><a NAME="E1:122_1_1_2_1_2_9_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font>
 ;<br/><div class="add"><a NAME="E2:122_1_1_2_1_2_9_1_2"/><b>then </b>
<font color="Maroon">P2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B4</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <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#T40" target="_self">Th40</a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:122_1_1_2_1_2_9_1_2"/><b>then </b>
<font color="Maroon">P2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B4</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E4:122_1_1_2_1_2_9_1_2"/><b>then </b><i><font color="Green">E372</font></i>: 
<font color="Maroon">P2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B4</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T33" target="_self">Th33</a>, , , <a class="ref" href="connsp_1.html#T38">CONNSP_1:38</a></i>;<br/><a NAME="E5:122_1_1_2_1_2_9_1_2"/>
<font color="Maroon">B3</font> is <a href="jordan2c.html#V1" target="_self">Bounded</a>
 <b>by </b><i><a class="ref" href="jordan2c.html#T40" target="_self">Th40</a>, </i>;<br/><b>hence </b><a NAME="E6:122_1_1_2_1_2_9_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan2c.html#T38" target="_self">Th38</a>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E29:122_1_1_2_1_2"/>
 <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">A</font> is <a href="jordan2c.html#V1" target="_self">Bounded</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:122_1_1"/>
 <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">A</font> is <a href="jordan2c.html#V1" target="_self">Bounded</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_2"><b>suppose </b></a><a NAME="E1:122_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><div class="add"><a NAME="E2:122_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="binop_2.html#K23">+</a> 1
 ;<br/><a NAME="E3:122_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:122_1_2"/><b>then </b><i><font color="Green">E373</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E5:122_1_2"/>
for <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>  holds <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="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>

<a NAME="E1:122_1_2_1"/>
<font color="Maroon">q2</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>
 
;<br/>
<a NAME="E2:122_1_2_1"/><b>then </b>
<font color="Maroon">q2</font> <a href="hidden.html#R2">in</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<a NAME="E3:122_1_2_1"/><b>then </b>
<font color="Maroon">q2</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font>
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:122_1_2_1"/><b>then </b>
<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> 0
 
<b>by </b><i><a class="ref" href="toprns_1.html#T24">TOPRNS_1:24</a></i>;<br/>
<b>hence </b><a NAME="E5:122_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="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/>


</div><b>end;</b></div><a NAME="E6:122_1_2"/><b>then </b>
for <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 <font color="Olive">q2</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> holds <br/><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="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><a NAME="E7:122_1_2"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> is <a href="jordan2c.html#V1" target="_self">Bounded</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:122_1_2"/>
 <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">A</font> is <a href="jordan2c.html#V1" target="_self">Bounded</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
