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

<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:122"/>
<font color="Maroon" title="c2">A</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:122"/><i><font color="Green" title="E108">A1</font></i>: 
for <font color="Olive" title="b1">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T40" target="_self" title="JORDAN2C:th.40">Th40</a></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" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 or <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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" title="E109">A2</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c4">a</font> = <font color="Maroon" title="c3">r</font>;<br/><b>reconsider </b><font color="Maroon" title="c5">P</font> = <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25" title="EUCLID:th.25">EUCLID:25</a></i>;<br/><a NAME="E3:122_1_1"/><i><font color="Green" title="E110">A3</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</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" title="c6">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:122_1_1_1"/><i><font color="Green" title="E111">A4</font></i>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 ;<br/>

<a NAME="E2:122_1_1_1"/><b>then </b><i><font color="Green" title="E112">A5</font></i>: 
( <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> &amp; not <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> </span>}</span>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">q0</font> = <font color="Maroon" title="c6">z</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="txt" href="jordan2c.html#E1:122_1_1_1"><i><font color="Green" title="E111">A4</font></i></a></i>;<br/>
<a NAME="E4:122_1_1_1"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c7">q0</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E2:122_1_1_1"><i><font color="Green" title="E112">A5</font></i></a></i>;<br/>
<a NAME="E5:122_1_1_1"/><b>then </b>
not <font color="Maroon" title="c7">q0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E3:122"><i><font color="Green" title="E108">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:122_1_1_1"/>
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:122_1_1"/><b>then </b><i><font color="Green" title="E111">A6</font></i>: 
 <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> <font color="Maroon" title="c5">P</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">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" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 or <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ;<br/><div class="add"><a NAME="E2:122_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c5">P</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>by </b><i><a class="ref" href="jordan2c.html#T61" target="_self" title="JORDAN2C:th.61">Th61</a></i>;<br/><a NAME="E3:122_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E112">A7</font></i>: 
 <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> <font color="Maroon" title="c5">P</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:122_1_1"><i><font color="Green" title="E111">A6</font></i></a>, <a class="ref" href="jordan2c.html#T15" target="_self" title="JORDAN2C:th.15">Th15</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" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">A</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E3:122_1_1_2_1_1_1"/><i><font color="Green" title="E113">A8</font></i>: 
( <font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">A</font> &amp; not <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c6">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> )
 <b>by </b><i><a class="ref" href="jordan2c.html#T40" target="_self" title="JORDAN2C:th.40">Th40</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:122_1_1_2_1_1_1"/><i><font color="Green" title="E114">A9</font></i>: 
( <font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">y</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B3</font> where <font color="Olive" title="b1">B3</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <font color="Olive" title="b1">B3</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c2">A</font> </span>}</span>  )
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:122_1_1_2_1_1_1"><i><font color="Green" title="E113">A8</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">B3</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E7:122_1_1_2_1_1_1"/><i><font color="Green" title="E115">A10</font></i>: 
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B3</font> &amp; <font color="Maroon" title="c8">B3</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E5:122_1_1_2_1_1_1"><i><font color="Green" title="E114">A9</font></i></a></i>;<br/><a NAME="E8:122_1_1_2_1_1_1"/>
<font color="Maroon" title="c8">B3</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="jordan2c.html#E7:122_1_1_2_1_1_1"><i><font color="Green" title="E115">A10</font></i></a>, <a class="ref" href="jordan2c.html#D3" target="_self" title="JORDAN2C:def.3">Def3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">B4</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E10:122_1_1_2_1_1_1"/><i><font color="Green" title="E116">A11</font></i>: 
( <font color="Maroon" title="c9">B4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B3</font> &amp; <font color="Maroon" title="c9">B4</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="connsp_1.html#D6" title="CONNSP_1:def.6">CONNSP_1:def 6</a></i>;<br/><a NAME="E11:122_1_1_2_1_1_1"/><i><font color="Green" title="E117">A12</font></i>: 
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 ;<br/><a NAME="E12:122_1_1_2_1_1_1"/>
for <font color="Olive" title="b1">q2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:122_1_1_2_1_1_1"><i><font color="Green" title="E113">A8</font></i></a></i>;<br/><a NAME="E13:122_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> &amp; not <font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q2</font> where <font color="Olive" title="b1">q2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> </span>}</span>  )
 <b>by </b><i><a class="txt" href="jordan2c.html#E11:122_1_1_2_1_1_1"><i><font color="Green" title="E117">A12</font></i></a>, <a class="ref" href="euclid.html#T25" title="EUCLID:th.25">EUCLID:25</a></i>;<br/><a NAME="E14:122_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E15:122_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">B4</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E5:122_1_1_2_1_1_1"><i><font color="Green" title="E114">A9</font></i></a>, <a class="txt" href="jordan2c.html#E7:122_1_1_2_1_1_1"><i><font color="Green" title="E115">A10</font></i></a>, <a class="txt" href="jordan2c.html#E10:122_1_1_2_1_1_1"><i><font color="Green" title="E116">A11</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E16:122_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">P</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c9">B4</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E17:122_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E118">A13</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">B4</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:122_1_1"><i><font color="Green" title="E111">A6</font></i></a>, <a class="txt" href="jordan2c.html#E3:122_1_1_2_1_1"><i><font color="Green" title="E112">A7</font></i></a>, <a class="txt" href="jordan2c.html#E10:122_1_1_2_1_1_1"><i><font color="Green" title="E116">A11</font></i></a>, <a class="ref" href="connsp_1.html#T38" title="CONNSP_1:th.38">CONNSP_1:38</a></i>;<br/><a NAME="E18:122_1_1_2_1_1_1"/>
<font color="Maroon" title="c8">B3</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E7:122_1_1_2_1_1_1"><i><font color="Green" title="E115">A10</font></i></a>, <a class="ref" href="jordan2c.html#D3" target="_self" title="JORDAN2C:def.3">Def3</a></i>;<br/><a NAME="E19:122_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">P</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E10:122_1_1_2_1_1_1"><i><font color="Green" title="E116">A11</font></i></a>, <a class="txt" href="jordan2c.html#E17:122_1_1_2_1_1_1"><i><font color="Green" title="E118">A13</font></i></a>, <a class="ref" href="jordan2c.html#T16" target="_self" title="JORDAN2C:th.16">Th16</a></i>;<br/><b>hence </b><a NAME="E20:122_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:122_1_1"><i><font color="Green" title="E109">A2</font></i></a>, <a class="ref" href="jordan2c.html#T62" target="_self" title="JORDAN2C:th.62">Th62</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" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">A</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">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" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2
 ;<br/><div class="add"><a NAME="E2:122_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 ;<br/><a NAME="E3:122_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:122_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E112">A14</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:122_1_1"><i><font color="Green" title="E109">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:122_1_1_2_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : for <font color="Olive" title="b2">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Olive" title="b2">r2</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span> holds <br/><font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">r</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">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" title="c6">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:122_1_1_2_1_2_1"/>
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : for <font color="Olive" title="b2">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Olive" title="b2">r2</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span> holds <br/><font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">r</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E3:122_1_1_2_1_2_1"/><i><font color="Green" title="E113">A15</font></i>: 
( <font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z</font> &amp; ( for <font color="Olive" title="b1">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Olive" title="b1">r2</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span> holds <br/><font color="Olive" title="b1">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">r</font> ) )
 ;<br/>
<b>thus </b><a NAME="E4:122_1_1_2_1_2_1"/>
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:122_1_1_2_1_2_1"><i><font color="Green" title="E113">A15</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c6">P1</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : for <font color="Olive" title="b2">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Olive" title="b2">r2</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span> holds <br/><font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">r</font> </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/><a NAME="E7:122_1_1_2_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : for <font color="Olive" title="b2">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Olive" title="b2">r2</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span> holds <br/><font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">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" title="c7">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:122_1_1_2_1_2_2"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : for <font color="Olive" title="b2">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Olive" title="b2">r2</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span> holds <br/><font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E3:122_1_1_2_1_2_2"/><i><font color="Green" title="E113">A16</font></i>: 
( <font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font> &amp; ( for <font color="Olive" title="b1">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Olive" title="b1">r2</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span> holds <br/><font color="Olive" title="b1">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font> ) )
 ;<br/>
<b>thus </b><a NAME="E4:122_1_1_2_1_2_2"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:122_1_1_2_1_2_2"><i><font color="Green" title="E113">A16</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c7">P2</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : for <font color="Olive" title="b2">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Olive" title="b2">r2</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span> holds <br/><font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font> </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/><a NAME="E9:122_1_1_2_1_2"/><i><font color="Green" title="E113">A17</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">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" title="c8">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:122_1_1_2_1_2_3"/><i><font color="Green" title="E114">A18</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 ;<br/>

<a NAME="E2:122_1_1_2_1_2_3"/><b>then </b><i><font color="Green" title="E115">A19</font></i>: 
( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> &amp; not <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> </span>}</span>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">q0</font> = <font color="Maroon" title="c8">z</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="txt" href="jordan2c.html#E1:122_1_1_2_1_2_3"><i><font color="Green" title="E114">A18</font></i></a></i>;<br/>
<a NAME="E4:122_1_1_2_1_2_3"/><i><font color="Green" title="E116">A20</font></i>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c9">q0</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E2:122_1_1_2_1_2_3"><i><font color="Green" title="E115">A19</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">r3</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E6:122_1_1_2_1_2_3"/><i><font color="Green" title="E117">A21</font></i>: 
<font color="Maroon" title="c9">q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c10">r3</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2"><i><font color="Green" title="E112">A14</font></i></a>, <a class="ref" href="jordan2b.html#T24" title="JORDAN2B:th.24">JORDAN2B:24</a></i>;<br/>
<a NAME="E7:122_1_1_2_1_2_3"/><i><font color="Green" title="E118">A22</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c10">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2_3"><i><font color="Green" title="E116">A20</font></i></a>, <a class="txt" href="jordan2c.html#E6:122_1_1_2_1_2_3"><i><font color="Green" title="E117">A21</font></i></a>, <a class="ref" href="jordan2c.html#T12" target="_self" title="JORDAN2C:th.12">Th12</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" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">r3</font> or <font color="Maroon" title="c10">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font> )
 </span><b>by </b><i><a class="txt" href="jordan2c.html#E7:122_1_1_2_1_2_3"><i><font color="Green" title="E118">A22</font></i></a>, <a class="ref" href="seq_2.html#T9" title="SEQ_2:th.9">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" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">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" title="b1">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c9">q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Olive" title="b1">r2</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span> holds <br/><font color="Olive" title="b1">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E6:122_1_1_2_1_2_3"><i><font color="Green" title="E117">A21</font></i></a>, <a class="ref" href="jordan2b.html#T29" title="JORDAN2B:th.29">JORDAN2B:29</a></i>;<br/><a NAME="E3:122_1_1_2_1_2_3_1_1"/><b>then </b>
<font color="Maroon" title="c9">q0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font>
 ;<br/><b>hence </b><a NAME="E4:122_1_1_2_1_2_3_1_1"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">P2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="c10">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">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" title="b1">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c9">q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Olive" title="b1">r2</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span> holds <br/><font color="Olive" title="b1">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E6:122_1_1_2_1_2_3"><i><font color="Green" title="E117">A21</font></i></a>, <a class="ref" href="jordan2b.html#T29" title="JORDAN2B:th.29">JORDAN2B:29</a></i>;<br/><a NAME="E3:122_1_1_2_1_2_3_1_2"/><b>then </b>
<font color="Maroon" title="c9">q0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P2</font>
 ;<br/><b>hence </b><a NAME="E4:122_1_1_2_1_2_3_1_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">P2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="c6">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">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" title="c8">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:122_1_1_2_1_2_4"/><i><font color="Green" title="E114">A23</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">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" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font> or <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P2</font> )
 </span><b>by </b><i><a class="txt" href="jordan2c.html#E1:122_1_1_2_1_2_4"><i><font color="Green" title="E114">A23</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c9">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">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" title="E115">A24</font></i>: 
( <font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z</font> &amp; ( for <font color="Olive" title="b1">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Olive" title="b1">r2</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span> holds <br/><font color="Olive" title="b1">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">r</font> ) )
 ;<br/><a NAME="E4:122_1_1_2_1_2_4_1_1"/><i><font color="Green" title="E116">A25</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:122_1_1_2_1_2_4_1_1"><i><font color="Green" title="E115">A24</font></i></a></i>;<br/><a NAME="E5:122_1_1_2_1_2_4_1_1"/>
for <font color="Olive" title="b1">q2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">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" title="c10">q2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">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" title="E117">A26</font></i>: 
<font color="Maroon" title="c10">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c11">r3</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:122_1_1_2_1_2_4_1_1_1"/><i><font color="Green" title="E118">A27</font></i>: 
<font color="Maroon" title="c10">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c11">r3</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2"><i><font color="Green" title="E112">A14</font></i></a>, <a class="ref" href="jordan2b.html#T24" title="JORDAN2B:th.24">JORDAN2B:24</a></i>;<br/>
<a NAME="E4:122_1_1_2_1_2_4_1_1_1"/><i><font color="Green" title="E119">A28</font></i>: 
<font color="Maroon" title="c11">r3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E3:122_1_1_2_1_2_4_1_1"><i><font color="Green" title="E115">A24</font></i></a>, <a class="txt" href="jordan2c.html#E1:122_1_1_2_1_2_4_1_1_1"><i><font color="Green" title="E117">A26</font></i></a>, <a class="txt" href="jordan2c.html#E3:122_1_1_2_1_2_4_1_1_1"><i><font color="Green" title="E118">A27</font></i></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" title="c3">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </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"/>
<font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:122_1_1_2_1_2_4_1_1_1_1_1_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c11">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</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" title="c3">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<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" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">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" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c11">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c11">r3</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2_4_1_1_1"><i><font color="Green" title="E119">A28</font></i></a>, <a class="ref" href="jordan2c.html#T1" target="_self" title="JORDAN2C:th.1">Th1</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" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c11">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2_4_1_1_1"><i><font color="Green" title="E119">A28</font></i></a>, <a class="ref" href="real_2.html#T109" title="REAL_2:th.109">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="E6:122_1_1_2_1_2_4_1_1_1"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c10">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:122_1_1_2_1_2_4_1_1_1"><i><font color="Green" title="E118">A27</font></i></a>, <a class="ref" href="jordan2c.html#T12" target="_self" title="JORDAN2C:th.12">Th12</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" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> &amp; not <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q2</font> where <font color="Olive" title="b1">q2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> </span>}</span>  )
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2_4_1_1"><i><font color="Green" title="E116">A25</font></i></a>, <a class="ref" href="euclid.html#T25" title="EUCLID:th.25">EUCLID:25</a></i>;<br/><b>hence </b><a NAME="E7:122_1_1_2_1_2_4_1_1"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">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" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c9">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">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" title="E115">A29</font></i>: 
( <font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z</font> &amp; ( for <font color="Olive" title="b1">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Olive" title="b1">r2</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span> holds <br/><font color="Olive" title="b1">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font> ) )
 ;<br/><a NAME="E4:122_1_1_2_1_2_4_1_2"/><i><font color="Green" title="E116">A30</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:122_1_1_2_1_2_4_1_2"><i><font color="Green" title="E115">A29</font></i></a></i>;<br/><a NAME="E5:122_1_1_2_1_2_4_1_2"/>
for <font color="Olive" title="b1">q2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">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" title="c10">q2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">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" title="E117">A31</font></i>: 
<font color="Maroon" title="c10">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c11">r3</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:122_1_1_2_1_2_4_1_2_1"/><i><font color="Green" title="E118">A32</font></i>: 
<font color="Maroon" title="c10">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c11">r3</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2"><i><font color="Green" title="E112">A14</font></i></a>, <a class="ref" href="jordan2b.html#T24" title="JORDAN2B:th.24">JORDAN2B:24</a></i>;<br/>
<a NAME="E4:122_1_1_2_1_2_4_1_2_1"/><i><font color="Green" title="E119">A33</font></i>: 
<font color="Maroon" title="c11">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E3:122_1_1_2_1_2_4_1_2"><i><font color="Green" title="E115">A29</font></i></a>, <a class="txt" href="jordan2c.html#E1:122_1_1_2_1_2_4_1_2_1"><i><font color="Green" title="E117">A31</font></i></a>, <a class="txt" href="jordan2c.html#E3:122_1_1_2_1_2_4_1_2_1"><i><font color="Green" title="E118">A32</font></i></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" title="c3">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </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"/>
<font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:122_1_1_2_1_2_4_1_2_1_1_1_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c11">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</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" title="c3">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<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" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c11">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2_4_1_2_1"><i><font color="Green" title="E119">A33</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">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="E6:122_1_1_2_1_2_4_1_2_1"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c10">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:122_1_1_2_1_2_4_1_2_1"><i><font color="Green" title="E118">A32</font></i></a>, <a class="ref" href="jordan2c.html#T12" target="_self" title="JORDAN2C:th.12">Th12</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" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> &amp; not <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q2</font> where <font color="Olive" title="b1">q2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> </span>}</span>  )
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2_4_1_2"><i><font color="Green" title="E116">A30</font></i></a>, <a class="ref" href="euclid.html#T25" title="EUCLID:th.25">EUCLID:25</a></i>;<br/><b>hence </b><a NAME="E7:122_1_1_2_1_2_4_1_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">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" title="E114">A34</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">P2</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E9:122_1_1_2_1_2"><i><font color="Green" title="E113">A17</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E12:122_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c6">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E13:122_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c6">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:122_1_1"><i><font color="Green" title="E110">A3</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E14:122_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E115">A35</font></i>: 
 <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> <font color="Maroon" title="c6">P1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/><a NAME="E15:122_1_1_2_1_2"/>
<font color="Maroon" title="c7">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E11:122_1_1_2_1_2"><i><font color="Green" title="E114">A34</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E16:122_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c7">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:122_1_1"><i><font color="Green" title="E110">A3</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E17:122_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E116">A36</font></i>: 
 <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> <font color="Maroon" title="c7">P2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">P2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/><a NAME="E18:122_1_1_2_1_2"/>
for <font color="Olive" title="b1">w1</font>, <font color="Olive" title="b2">w2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font> &amp; <font color="Olive" title="b2">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font> holds <br/> <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Olive" title="b1">w1</font>,<font color="Olive" title="b2">w2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">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" title="c8">w1</font>, <font color="Maroon" title="c9">w2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:122_1_1_2_1_2_5"/><i><font color="Green" title="E117">A37</font></i>: 
( <font color="Maroon" title="c8">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font> &amp; <font color="Maroon" title="c9">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">q1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E3:122_1_1_2_1_2_5"/><i><font color="Green" title="E118">A38</font></i>: 
( <font color="Maroon" title="c10">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">w1</font> &amp; ( for <font color="Olive" title="b1">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c10">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Olive" title="b1">r2</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span> holds <br/><font color="Olive" title="b1">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">r</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon" title="c11">q2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E5:122_1_1_2_1_2_5"/><i><font color="Green" title="E119">A39</font></i>: 
( <font color="Maroon" title="c11">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">w2</font> &amp; ( for <font color="Olive" title="b1">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c11">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Olive" title="b1">r2</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span> holds <br/><font color="Olive" title="b1">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">r</font> ) )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:122_1_1_2_1_2_5"><i><font color="Green" title="E117">A37</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">r3</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E7:122_1_1_2_1_2_5"/><i><font color="Green" title="E120">A40</font></i>: 
<font color="Maroon" title="c8">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c12">r3</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2"><i><font color="Green" title="E112">A14</font></i></a>, <a class="ref" href="jordan2b.html#T24" title="JORDAN2B:th.24">JORDAN2B:24</a></i>;<br/>
<a NAME="E8:122_1_1_2_1_2_5"/><i><font color="Green" title="E121">A41</font></i>: 
<font color="Maroon" title="c12">r3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E3:122_1_1_2_1_2_5"><i><font color="Green" title="E118">A38</font></i></a>, <a class="txt" href="jordan2c.html#E7:122_1_1_2_1_2_5"><i><font color="Green" title="E120">A40</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">r4</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E10:122_1_1_2_1_2_5"/><i><font color="Green" title="E122">A42</font></i>: 
<font color="Maroon" title="c9">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c13">r4</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2"><i><font color="Green" title="E112">A14</font></i></a>, <a class="ref" href="jordan2b.html#T24" title="JORDAN2B:th.24">JORDAN2B:24</a></i>;<br/>
<a NAME="E11:122_1_1_2_1_2_5"/><i><font color="Green" title="E123">A43</font></i>: 
<font color="Maroon" title="c13">r4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E5:122_1_1_2_1_2_5"><i><font color="Green" title="E119">A39</font></i></a>, <a class="txt" href="jordan2c.html#E10:122_1_1_2_1_2_5"><i><font color="Green" title="E122">A42</font></i></a></i>;<br/>
<b>thus </b><a NAME="E12:122_1_1_2_1_2_5"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c8">w1</font>,<font color="Maroon" title="c9">w2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">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" title="c14">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:122_1_1_2_1_2_5_1"/>
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c8">w1</font>,<font color="Maroon" title="c9">w2</font>
 ;<br/>

<a NAME="E2:122_1_1_2_1_2_5_1"/><b>then </b>
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">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" title="BINOP_2:func.10">-</a> <font color="Olive" title="b1">r2</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c8">w1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">r2</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c9">w2</font></span>)</span></span>)</span> where <font color="Olive" title="b1">r2</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r2</font> &amp; <font color="Olive" title="b1">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">r2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:122_1_1_2_1_2_5_1"/><i><font color="Green" title="E124">A44</font></i>: 
( <font color="Maroon" title="c14">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c15">r2</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c8">w1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">r2</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c9">w2</font></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">r2</font> &amp; <font color="Maroon" title="c15">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/>
<a NAME="E5:122_1_1_2_1_2_5_1"/><i><font color="Green" title="E125">A45</font></i>: 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c15">r2</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c8">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c15">r2</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c12">r3</font></span>)</span></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E7:122_1_1_2_1_2_5"><i><font color="Green" title="E120">A40</font></i></a>, <a class="ref" href="jordan2b.html#T26" title="JORDAN2B:th.26">JORDAN2B:26</a></i>;<br/>
<a NAME="E6:122_1_1_2_1_2_5_1"/>
<font color="Maroon" title="c15">r2</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c9">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">r2</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c13">r4</font></span>)</span></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E10:122_1_1_2_1_2_5"><i><font color="Green" title="E122">A42</font></i></a>, <a class="ref" href="jordan2b.html#T26" title="JORDAN2B:th.26">JORDAN2B:26</a></i>;<br/>
<a NAME="E7:122_1_1_2_1_2_5_1"/><b>then </b><i><font color="Green" title="E126">A46</font></i>: 
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</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" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c15">r2</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c12">r3</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">r2</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c13">r4</font></span>)</span></span>)</span></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2_5_1"><i><font color="Green" title="E124">A44</font></i></a>, <a class="txt" href="jordan2c.html#E5:122_1_1_2_1_2_5_1"><i><font color="Green" title="E125">A45</font></i></a>, <a class="ref" href="jordan2b.html#T27" title="JORDAN2B:th.27">JORDAN2B:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">q4</font> = <font color="Maroon" title="c14">z</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2_5_1"><i><font color="Green" title="E124">A44</font></i></a></i>;<br/>
<a NAME="E9:122_1_1_2_1_2_5_1"/>
for <font color="Olive" title="b1">r5</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c16">q4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Olive" title="b1">r5</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span> holds <br/><font color="Olive" title="b1">r5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">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" title="c17">r5</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:122_1_1_2_1_2_5_1_1"/>
<font color="Maroon" title="c16">q4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Maroon" title="c17">r5</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span>
 ;<br/>

<a NAME="E2:122_1_1_2_1_2_5_1_1"/><b>then </b><i><font color="Green" title="E127">A47</font></i>: 
<font color="Maroon" title="c17">r5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c15">r2</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c12">r3</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">r2</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c13">r4</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E7:122_1_1_2_1_2_5_1"><i><font color="Green" title="E126">A46</font></i></a>, <a class="ref" href="jordan2b.html#T29" title="JORDAN2B:th.29">JORDAN2B:29</a></i>;<br/>
<a NAME="E3:122_1_1_2_1_2_5_1_1"/>
1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c15">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2_5_1"><i><font color="Green" title="E124">A44</font></i></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">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" title="E128">A48</font></i>: 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c15">r2</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c12">r3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c15">r2</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E8:122_1_1_2_1_2_5"><i><font color="Green" title="E121">A41</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<a NAME="E5:122_1_1_2_1_2_5_1_1"/><i><font color="Green" title="E129">A49</font></i>: 
<font color="Maroon" title="c15">r2</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c13">r4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">r2</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E11:122_1_1_2_1_2_5"><i><font color="Green" title="E123">A43</font></i></a>, <a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2_5_1"><i><font color="Green" title="E124">A44</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">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" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c15">r2</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">r2</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">r</font>
 
;<br/>
<b>hence </b><a NAME="E7:122_1_1_2_1_2_5_1_1"/>
<font color="Maroon" title="c17">r5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E2:122_1_1_2_1_2_5_1_1"><i><font color="Green" title="E127">A47</font></i></a>, <a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2_5_1_1"><i><font color="Green" title="E128">A48</font></i></a>, <a class="txt" href="jordan2c.html#E5:122_1_1_2_1_2_5_1_1"><i><font color="Green" title="E129">A49</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">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" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">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" title="c6">P1</font> is <a href="jordan1.html#V1" title="JORDAN1:attr.1">convex</a>
 <b>by </b><i><a class="ref" href="jordan1.html#D1" title="JORDAN1:def.1">JORDAN1:def 1</a></i>;<br/><a NAME="E20:122_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E117">A50</font></i>: 
<font color="Maroon" title="c6">P1</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>by </b><i><a class="ref" href="jordan1.html#T9" title="JORDAN1:th.9">JORDAN1:9</a></i>;<br/><a NAME="E21:122_1_1_2_1_2"/>
for <font color="Olive" title="b1">w1</font>, <font color="Olive" title="b2">w2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P2</font> &amp; <font color="Olive" title="b2">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P2</font> holds <br/> <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Olive" title="b1">w1</font>,<font color="Olive" title="b2">w2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">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" title="c8">w1</font>, <font color="Maroon" title="c9">w2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:122_1_1_2_1_2_6"/><i><font color="Green" title="E118">A51</font></i>: 
( <font color="Maroon" title="c8">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P2</font> &amp; <font color="Maroon" title="c9">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P2</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">q1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E3:122_1_1_2_1_2_6"/><i><font color="Green" title="E119">A52</font></i>: 
( <font color="Maroon" title="c10">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">w1</font> &amp; ( for <font color="Olive" title="b1">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c10">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Olive" title="b1">r2</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span> holds <br/><font color="Olive" title="b1">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon" title="c11">q2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E5:122_1_1_2_1_2_6"/><i><font color="Green" title="E120">A53</font></i>: 
( <font color="Maroon" title="c11">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">w2</font> &amp; ( for <font color="Olive" title="b1">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c11">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Olive" title="b1">r2</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span> holds <br/><font color="Olive" title="b1">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font> ) )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:122_1_1_2_1_2_6"><i><font color="Green" title="E118">A51</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">r3</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E7:122_1_1_2_1_2_6"/><i><font color="Green" title="E121">A54</font></i>: 
<font color="Maroon" title="c8">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c12">r3</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2"><i><font color="Green" title="E112">A14</font></i></a>, <a class="ref" href="jordan2b.html#T24" title="JORDAN2B:th.24">JORDAN2B:24</a></i>;<br/>
<a NAME="E8:122_1_1_2_1_2_6"/><i><font color="Green" title="E122">A55</font></i>: 
<font color="Maroon" title="c12">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E3:122_1_1_2_1_2_6"><i><font color="Green" title="E119">A52</font></i></a>, <a class="txt" href="jordan2c.html#E7:122_1_1_2_1_2_6"><i><font color="Green" title="E121">A54</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">r4</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E10:122_1_1_2_1_2_6"/><i><font color="Green" title="E123">A56</font></i>: 
<font color="Maroon" title="c9">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c13">r4</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2"><i><font color="Green" title="E112">A14</font></i></a>, <a class="ref" href="jordan2b.html#T24" title="JORDAN2B:th.24">JORDAN2B:24</a></i>;<br/>
<a NAME="E11:122_1_1_2_1_2_6"/><i><font color="Green" title="E124">A57</font></i>: 
<font color="Maroon" title="c13">r4</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E5:122_1_1_2_1_2_6"><i><font color="Green" title="E120">A53</font></i></a>, <a class="txt" href="jordan2c.html#E10:122_1_1_2_1_2_6"><i><font color="Green" title="E123">A56</font></i></a></i>;<br/>
<b>thus </b><a NAME="E12:122_1_1_2_1_2_6"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c8">w1</font>,<font color="Maroon" title="c9">w2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">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" title="c14">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:122_1_1_2_1_2_6_1"/>
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c8">w1</font>,<font color="Maroon" title="c9">w2</font>
 ;<br/>

<a NAME="E2:122_1_1_2_1_2_6_1"/><b>then </b>
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">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" title="BINOP_2:func.10">-</a> <font color="Olive" title="b1">r2</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c8">w1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">r2</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c9">w2</font></span>)</span></span>)</span> where <font color="Olive" title="b1">r2</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r2</font> &amp; <font color="Olive" title="b1">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">r2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:122_1_1_2_1_2_6_1"/><i><font color="Green" title="E125">A58</font></i>: 
( <font color="Maroon" title="c14">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c15">r2</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c8">w1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">r2</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c9">w2</font></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">r2</font> &amp; <font color="Maroon" title="c15">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/>
<a NAME="E5:122_1_1_2_1_2_6_1"/><i><font color="Green" title="E126">A59</font></i>: 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c15">r2</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c8">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c15">r2</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c12">r3</font></span>)</span></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E7:122_1_1_2_1_2_6"><i><font color="Green" title="E121">A54</font></i></a>, <a class="ref" href="jordan2b.html#T26" title="JORDAN2B:th.26">JORDAN2B:26</a></i>;<br/>
<a NAME="E6:122_1_1_2_1_2_6_1"/>
<font color="Maroon" title="c15">r2</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c9">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">r2</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c13">r4</font></span>)</span></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E10:122_1_1_2_1_2_6"><i><font color="Green" title="E123">A56</font></i></a>, <a class="ref" href="jordan2b.html#T26" title="JORDAN2B:th.26">JORDAN2B:26</a></i>;<br/>
<a NAME="E7:122_1_1_2_1_2_6_1"/><b>then </b><i><font color="Green" title="E127">A60</font></i>: 
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</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" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c15">r2</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c12">r3</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">r2</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c13">r4</font></span>)</span></span>)</span></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2_6_1"><i><font color="Green" title="E125">A58</font></i></a>, <a class="txt" href="jordan2c.html#E5:122_1_1_2_1_2_6_1"><i><font color="Green" title="E126">A59</font></i></a>, <a class="ref" href="jordan2b.html#T27" title="JORDAN2B:th.27">JORDAN2B:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">q4</font> = <font color="Maroon" title="c14">z</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2_6_1"><i><font color="Green" title="E125">A58</font></i></a></i>;<br/>
<a NAME="E9:122_1_1_2_1_2_6_1"/>
for <font color="Olive" title="b1">r5</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c16">q4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Olive" title="b1">r5</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span> holds <br/><font color="Olive" title="b1">r5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">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" title="c17">r5</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:122_1_1_2_1_2_6_1_1"/>
<font color="Maroon" title="c16">q4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Maroon" title="c17">r5</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span>
 ;<br/>

<a NAME="E2:122_1_1_2_1_2_6_1_1"/><b>then </b><i><font color="Green" title="E128">A61</font></i>: 
<font color="Maroon" title="c17">r5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c15">r2</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c12">r3</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">r2</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c13">r4</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E7:122_1_1_2_1_2_6_1"><i><font color="Green" title="E127">A60</font></i></a>, <a class="ref" href="jordan2b.html#T29" title="JORDAN2B:th.29">JORDAN2B:29</a></i>;<br/>
<a NAME="E3:122_1_1_2_1_2_6_1_1"/>
1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c15">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2_6_1"><i><font color="Green" title="E125">A58</font></i></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">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" title="E129">A62</font></i>: 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c15">r2</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c12">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c15">r2</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E8:122_1_1_2_1_2_6"><i><font color="Green" title="E122">A55</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<a NAME="E5:122_1_1_2_1_2_6_1_1"/><i><font color="Green" title="E130">A63</font></i>: 
<font color="Maroon" title="c15">r2</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c13">r4</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">r2</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E11:122_1_1_2_1_2_6"><i><font color="Green" title="E124">A57</font></i></a>, <a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2_6_1"><i><font color="Green" title="E125">A58</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">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" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c15">r2</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">r2</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font>
 
;<br/>
<b>hence </b><a NAME="E7:122_1_1_2_1_2_6_1_1"/>
<font color="Maroon" title="c17">r5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E2:122_1_1_2_1_2_6_1_1"><i><font color="Green" title="E128">A61</font></i></a>, <a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2_6_1_1"><i><font color="Green" title="E129">A62</font></i></a>, <a class="txt" href="jordan2c.html#E5:122_1_1_2_1_2_6_1_1"><i><font color="Green" title="E130">A63</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">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" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">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" title="c7">P2</font> is <a href="jordan1.html#V1" title="JORDAN1:attr.1">convex</a>
 <b>by </b><i><a class="ref" href="jordan1.html#D1" title="JORDAN1:def.1">JORDAN1:def 1</a></i>;<br/><a NAME="E23:122_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E118">A64</font></i>: 
<font color="Maroon" title="c7">P2</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>by </b><i><a class="ref" href="jordan1.html#T9" title="JORDAN1:th.9">JORDAN1:9</a></i>;<br/><div><i><font color="Green" title="E119">A65</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" title="c6">P1</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 ;<br/><b>then consider </b><font color="Maroon" title="c8">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:122_1_1_2_1_2_7"/><i><font color="Green" title="E120">A66</font></i>: 
for <font color="Olive" title="b1">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T40" target="_self" title="JORDAN2C:th.40">Th40</a></i>;<br/><b>set </b><font color="Maroon" title="c9">p3</font> = <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span>;<br/><a NAME="E4:122_1_1_2_1_2_7"/>
for <font color="Olive" title="b1">r5</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Olive" title="b1">r5</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span> holds <br/><font color="Olive" title="b1">r5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">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" title="c10">r5</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">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" title="JORDAN2B:func.2">|[</a><span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Maroon" title="c10">r5</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span>
 ;<br/>

<a NAME="E2:122_1_1_2_1_2_7_1"/><b>then </b><i><font color="Green" title="E121">A67</font></i>: 
<font color="Maroon" title="c10">r5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan2b.html#T29" title="JORDAN2B:th.29">JORDAN2B:29</a></i>;<br/>
<a NAME="E3:122_1_1_2_1_2_7_1"/>
<font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#T11" title="ABSVALUE:th.11">ABSVALUE:11</a></i>;<br/>
<a NAME="E4:122_1_1_2_1_2_7_1"/><b>then </b><i><font color="Green" title="E122">A68</font></i>: 
 <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/>
<a NAME="E5:122_1_1_2_1_2_7_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/>
<a NAME="E6:122_1_1_2_1_2_7_1"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="real_2.html#T173" title="REAL_2:th.173">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" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/>
<b>hence </b><a NAME="E8:122_1_1_2_1_2_7_1"/>
<font color="Maroon" title="c10">r5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E2:122_1_1_2_1_2_7_1"><i><font color="Green" title="E121">A67</font></i></a>, <a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2_7_1"><i><font color="Green" title="E122">A68</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">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" title="JORDAN2B:func.2">|[</a><span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2"><i><font color="Green" title="E112">A14</font></i></a></i>;<br/><a NAME="E6:122_1_1_2_1_2_7"/><b>then </b><i><font color="Green" title="E121">A69</font></i>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:122_1_1_2_1_2_7"><i><font color="Green" title="E120">A66</font></i></a></i>;<br/><i><font color="Green" title="E122">A70</font></i>: <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> = 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan2c.html#T12" target="_self" title="JORDAN2C:th.12">Th12</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></i>
;<br/><a NAME="E8:122_1_1_2_1_2_7"/><i><font color="Green" title="E123">A71</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font>
 <b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/><a NAME="E9:122_1_1_2_1_2_7"/><i><font color="Green" title="E124">A72</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/><a NAME="E10:122_1_1_2_1_2_7"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E8:122_1_1_2_1_2_7"><i><font color="Green" title="E123">A71</font></i></a>, <a class="ref" href="xreal_1.html#T35" title="XREAL_1:th.35">XREAL_1:35</a></i>;<br/><a NAME="E11:122_1_1_2_1_2_7"/><b>then </b><i><font color="Green" title="E125">A73</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E12:122_1_1_2_1_2_7"/><i><font color="Green" title="E126">A74</font></i>: 
<font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font>
 <b>by </b><i><a class="ref" href="absvalue.html#T11" title="ABSVALUE:th.11">ABSVALUE:11</a></i>;<br/><a NAME="E13:122_1_1_2_1_2_7"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E9:122_1_1_2_1_2_7"><i><font color="Green" title="E124">A72</font></i></a>, <a class="ref" href="real_2.html#T173" title="REAL_2:th.173">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="txt" href="jordan2c.html#E6:122_1_1_2_1_2_7"><i><font color="Green" title="E121">A69</font></i></a>, <a class="txt" href="jordan2c.html#E7:122_1_1_2_1_2_7"><i><font color="Green" title="E122">A70</font></i></a>, <a class="txt" href="jordan2c.html#E11:122_1_1_2_1_2_7"><i><font color="Green" title="E125">A73</font></i></a>, <a class="txt" href="jordan2c.html#E12:122_1_1_2_1_2_7"><i><font color="Green" title="E126">A74</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E120">A75</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" title="c7">P2</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 ;<br/><b>then consider </b><font color="Maroon" title="c8">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:122_1_1_2_1_2_8"/><i><font color="Green" title="E121">A76</font></i>: 
for <font color="Olive" title="b1">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P2</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T40" target="_self" title="JORDAN2C:th.40">Th40</a></i>;<br/><b>set </b><font color="Maroon" title="c9">p3</font> = <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span>;<br/><a NAME="E4:122_1_1_2_1_2_8"/>
for <font color="Olive" title="b1">r5</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Olive" title="b1">r5</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span> holds <br/><font color="Olive" title="b1">r5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">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" title="c10">r5</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">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" title="JORDAN2B:func.2">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><font color="Maroon" title="c10">r5</font></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span>
 ;<br/>

<a NAME="E2:122_1_1_2_1_2_8_1"/><b>then </b><i><font color="Green" title="E122">A77</font></i>: 
<font color="Maroon" title="c10">r5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan2b.html#T29" title="JORDAN2B:th.29">JORDAN2B:29</a></i>;<br/>
<a NAME="E3:122_1_1_2_1_2_8_1"/><i><font color="Green" title="E123">A78</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#T11" title="ABSVALUE:th.11">ABSVALUE:11</a></i>;<br/>
<a NAME="E4:122_1_1_2_1_2_8_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/>
<a NAME="E5:122_1_1_2_1_2_8_1"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="real_2.html#T173" title="REAL_2:th.173">REAL_2:173</a></i>;<br/>
<b>hence </b><a NAME="E6:122_1_1_2_1_2_8_1"/>
<font color="Maroon" title="c10">r5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E2:122_1_1_2_1_2_8_1"><i><font color="Green" title="E122">A77</font></i></a>, <a class="txt" href="jordan2c.html#E3:122_1_1_2_1_2_8_1"><i><font color="Green" title="E123">A78</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">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" title="JORDAN2B:func.2">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P2</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2"><i><font color="Green" title="E112">A14</font></i></a></i>;<br/><a NAME="E6:122_1_1_2_1_2_8"/><b>then </b><i><font color="Green" title="E122">A79</font></i>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:122_1_1_2_1_2_8"><i><font color="Green" title="E121">A76</font></i></a></i>;<br/><a NAME="E7:122_1_1_2_1_2_8"/><i><font color="Green" title="E123">A80</font></i>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2"><a href="jordan2b.html#K2" title="JORDAN2B:func.2">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span><a href="jordan2b.html#K2" title="JORDAN2B:func.2">]|</a></span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T12" target="_self" title="JORDAN2C:th.12">Th12</a></i>;<br/><a NAME="E8:122_1_1_2_1_2_8"/><i><font color="Green" title="E124">A81</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font>
 <b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/><a NAME="E9:122_1_1_2_1_2_8"/><i><font color="Green" title="E125">A82</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/><a NAME="E10:122_1_1_2_1_2_8"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E8:122_1_1_2_1_2_8"><i><font color="Green" title="E124">A81</font></i></a>, <a class="ref" href="xreal_1.html#T35" title="XREAL_1:th.35">XREAL_1:35</a></i>;<br/><a NAME="E11:122_1_1_2_1_2_8"/><b>then </b><i><font color="Green" title="E126">A83</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E12:122_1_1_2_1_2_8"/><i><font color="Green" title="E127">A84</font></i>: 
<font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font>
 <b>by </b><i><a class="ref" href="absvalue.html#T11" title="ABSVALUE:th.11">ABSVALUE:11</a></i>;<br/><a NAME="E13:122_1_1_2_1_2_8"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c3">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E9:122_1_1_2_1_2_8"><i><font color="Green" title="E125">A82</font></i></a>, <a class="ref" href="real_2.html#T173" title="REAL_2:th.173">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="txt" href="jordan2c.html#E6:122_1_1_2_1_2_8"><i><font color="Green" title="E122">A79</font></i></a>, <a class="txt" href="jordan2c.html#E7:122_1_1_2_1_2_8"><i><font color="Green" title="E123">A80</font></i></a>, <a class="txt" href="jordan2c.html#E11:122_1_1_2_1_2_8"><i><font color="Green" title="E126">A83</font></i></a>, <a class="txt" href="jordan2c.html#E12:122_1_1_2_1_2_8"><i><font color="Green" title="E127">A84</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><a NAME="E26:122_1_1_2_1_2"/><i><font color="Green" title="E121">A85</font></i>: 
 <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> <font color="Maroon" title="c6">P1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E14:122_1_1_2_1_2"><i><font color="Green" title="E115">A35</font></i></a>, <a class="txt" href="jordan2c.html#E20:122_1_1_2_1_2"><i><font color="Green" title="E117">A50</font></i></a>, <a class="ref" href="jordan2c.html#T15" target="_self" title="JORDAN2C:th.15">Th15</a></i>;<br/><a NAME="E27:122_1_1_2_1_2"/><i><font color="Green" title="E122">A86</font></i>: 
 <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> <font color="Maroon" title="c7">P2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E17:122_1_1_2_1_2"><i><font color="Green" title="E116">A36</font></i></a>, <a class="txt" href="jordan2c.html#E23:122_1_1_2_1_2"><i><font color="Green" title="E118">A64</font></i></a>, <a class="ref" href="jordan2c.html#T15" target="_self" title="JORDAN2C:th.15">Th15</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" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">A</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 ;<br/><b>then consider </b><font color="Maroon" title="c8">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E3:122_1_1_2_1_2_9"/><i><font color="Green" title="E123">A87</font></i>: 
( <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">A</font> &amp; not <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c8">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> )
 <b>by </b><i><a class="ref" href="jordan2c.html#T40" target="_self" title="JORDAN2C:th.40">Th40</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:122_1_1_2_1_2_9"/><i><font color="Green" title="E124">A88</font></i>: 
( <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B3</font> where <font color="Olive" title="b1">B3</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <font color="Olive" title="b1">B3</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c2">A</font> </span>}</span>  )
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:122_1_1_2_1_2_9"><i><font color="Green" title="E123">A87</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">B3</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E7:122_1_1_2_1_2_9"/><i><font color="Green" title="E125">A89</font></i>: 
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B3</font> &amp; <font color="Maroon" title="c10">B3</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E5:122_1_1_2_1_2_9"><i><font color="Green" title="E124">A88</font></i></a></i>;<br/><a NAME="E8:122_1_1_2_1_2_9"/>
<font color="Maroon" title="c10">B3</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="jordan2c.html#E7:122_1_1_2_1_2_9"><i><font color="Green" title="E125">A89</font></i></a>, <a class="ref" href="jordan2c.html#D3" target="_self" title="JORDAN2C:def.3">Def3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">B4</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E10:122_1_1_2_1_2_9"/><i><font color="Green" title="E126">A90</font></i>: 
( <font color="Maroon" title="c11">B4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B3</font> &amp; <font color="Maroon" title="c11">B4</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="connsp_1.html#D6" title="CONNSP_1:def.6">CONNSP_1:def 6</a></i>;<br/><a NAME="E11:122_1_1_2_1_2_9"/><i><font color="Green" title="E127">A91</font></i>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 ;<br/><a NAME="E12:122_1_1_2_1_2_9"/>
for <font color="Olive" title="b1">q2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:122_1_1_2_1_2_9"><i><font color="Green" title="E123">A87</font></i></a></i>;<br/><a NAME="E13:122_1_1_2_1_2_9"/><b>then </b>
( <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> &amp; not <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q2</font> where <font color="Olive" title="b1">q2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> </span>}</span>  )
 <b>by </b><i><a class="txt" href="jordan2c.html#E11:122_1_1_2_1_2_9"><i><font color="Green" title="E127">A91</font></i></a>, <a class="ref" href="euclid.html#T25" title="EUCLID:th.25">EUCLID:25</a></i>;<br/><a NAME="E14:122_1_1_2_1_2_9"/><b>then </b><i><font color="Green" title="E128">A92</font></i>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">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" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font> or <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P2</font> )
 </span><b>by </b><i><a class="txt" href="jordan2c.html#E9:122_1_1_2_1_2"><i><font color="Green" title="E113">A17</font></i></a>, <a class="txt" href="jordan2c.html#E14:122_1_1_2_1_2_9"><i><font color="Green" title="E128">A92</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font>
 ;<br/><div class="add"><a NAME="E2:122_1_1_2_1_2_9_1_1"/><b>then </b>
<font color="Maroon" title="c6">P1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c11">B4</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E5:122_1_1_2_1_2_9"><i><font color="Green" title="E124">A88</font></i></a>, <a class="txt" href="jordan2c.html#E7:122_1_1_2_1_2_9"><i><font color="Green" title="E125">A89</font></i></a>, <a class="txt" href="jordan2c.html#E10:122_1_1_2_1_2_9"><i><font color="Green" title="E126">A90</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">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" title="c6">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c11">B4</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">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" title="E129">A93</font></i>: 
<font color="Maroon" title="c6">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">B4</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E14:122_1_1_2_1_2"><i><font color="Green" title="E115">A35</font></i></a>, <a class="txt" href="jordan2c.html#E26:122_1_1_2_1_2"><i><font color="Green" title="E121">A85</font></i></a>, <a class="txt" href="jordan2c.html#E10:122_1_1_2_1_2_9"><i><font color="Green" title="E126">A90</font></i></a>, <a class="ref" href="connsp_1.html#T38" title="CONNSP_1:th.38">CONNSP_1:38</a></i>;<br/><a NAME="E5:122_1_1_2_1_2_9_1_1"/>
<font color="Maroon" title="c10">B3</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E7:122_1_1_2_1_2_9"><i><font color="Green" title="E125">A89</font></i></a>, <a class="ref" href="jordan2c.html#D3" target="_self" title="JORDAN2C:def.3">Def3</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="txt" href="jordan2c.html#E24:122_1_1_2_1_2"><i><font color="Green" title="E119">A65</font></i></a>, <a class="txt" href="jordan2c.html#E10:122_1_1_2_1_2_9"><i><font color="Green" title="E126">A90</font></i></a>, <a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2_9_1_1"><i><font color="Green" title="E129">A93</font></i></a>, <a class="ref" href="jordan2c.html#T16" target="_self" title="JORDAN2C:th.16">Th16</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" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P2</font>
 ;<br/><div class="add"><a NAME="E2:122_1_1_2_1_2_9_1_2"/><b>then </b>
<font color="Maroon" title="c7">P2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c11">B4</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E5:122_1_1_2_1_2_9"><i><font color="Green" title="E124">A88</font></i></a>, <a class="txt" href="jordan2c.html#E7:122_1_1_2_1_2_9"><i><font color="Green" title="E125">A89</font></i></a>, <a class="txt" href="jordan2c.html#E10:122_1_1_2_1_2_9"><i><font color="Green" title="E126">A90</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">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" title="c7">P2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c11">B4</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">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" title="E129">A94</font></i>: 
<font color="Maroon" title="c7">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">B4</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E17:122_1_1_2_1_2"><i><font color="Green" title="E116">A36</font></i></a>, <a class="txt" href="jordan2c.html#E27:122_1_1_2_1_2"><i><font color="Green" title="E122">A86</font></i></a>, <a class="txt" href="jordan2c.html#E10:122_1_1_2_1_2_9"><i><font color="Green" title="E126">A90</font></i></a>, <a class="ref" href="connsp_1.html#T38" title="CONNSP_1:th.38">CONNSP_1:38</a></i>;<br/><a NAME="E5:122_1_1_2_1_2_9_1_2"/>
<font color="Maroon" title="c10">B3</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 <b>by </b><i><a class="txt" href="jordan2c.html#E7:122_1_1_2_1_2_9"><i><font color="Green" title="E125">A89</font></i></a>, <a class="ref" href="jordan2c.html#D3" target="_self" title="JORDAN2C:def.3">Def3</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="txt" href="jordan2c.html#E25:122_1_1_2_1_2"><i><font color="Green" title="E120">A75</font></i></a>, <a class="txt" href="jordan2c.html#E10:122_1_1_2_1_2_9"><i><font color="Green" title="E126">A90</font></i></a>, <a class="txt" href="jordan2c.html#E4:122_1_1_2_1_2_9_1_2"><i><font color="Green" title="E129">A94</font></i></a>, <a class="ref" href="jordan2c.html#T16" target="_self" title="JORDAN2C:th.16">Th16</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" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">A</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">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" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">A</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">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" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ;<br/><div class="add"><a NAME="E2:122_1_2"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 ;<br/><a NAME="E3:122_1_2"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:122_1_2"/><b>then </b><i><font color="Green" title="E109">A95</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E5:122_1_2"/>
for <font color="Olive" title="b1">q2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  holds <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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" title="c4">q2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>

<a NAME="E1:122_1_2_1"/>
<font color="Maroon" title="c4">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
;<br/>
<a NAME="E2:122_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T25" title="EUCLID:th.25">EUCLID:25</a></i>;<br/>
<a NAME="E3:122_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E4:122_1_2"><i><font color="Green" title="E109">A95</font></i></a>, <a class="ref" href="jordan2c.html#T113" target="_self" title="JORDAN2C:th.113">Th113</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:122_1_2_1"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c4">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <b>by </b><i><a class="ref" href="toprns_1.html#T24" title="TOPRNS_1:th.24">TOPRNS_1:24</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:122_1_2"/><b>then </b>
for <font color="Olive" title="b1">q2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ;<br/><a NAME="E7:122_1_2"/><b>then </b>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 <b>by </b><i><a class="ref" href="jordan2c.html#T40" target="_self" title="JORDAN2C:th.40">Th40</a></i>;<br/><b>hence </b><a NAME="E8:122_1_2"/>
 <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">A</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 <b>by </b><i><a class="ref" href="jordan2c.html#T16" target="_self" title="JORDAN2C:th.16">Th16</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
