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

<b>set </b><font color="Maroon">X</font> =  <a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">a</font>;<br/>
<a NAME="E1:11_2_1"/>
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">a</font></span>)</span> <a href="subset_1.html#K3">`</a>  holds <br/> ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">p</font>,<font color="Olive">r</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">a</font></span>)</span> <a href="subset_1.html#K3">`</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:11_2_1_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">a</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E3:11_2_1_1"/><i><font color="Green">E85</font></i>: 
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="yellow_6.html#T10">YELLOW_6:10</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_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="E1:11_2_1_1_1"/>
( <font color="Maroon">x</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> <a href="euclid.html#K21">`1</a>  or <font color="Maroon">x</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> <a href="euclid.html#K22">`2</a>  )
 </span><b>by </b><i>, <a class="ref" href="topreal1.html#D14">TOPREAL1:def 14</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:11_2_1_1_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">x</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> <a href="euclid.html#K21">`1</a> 
 ;<br/><div class="add"><b>take </b><font color="Maroon">r</font> =  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>;<br/><a NAME="E2:11_2_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:11_2_1_1_1_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T133">COMPLEX1:133</a></i>;<br/><b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/><b>assume </b><a NAME="E4:11_2_1_1_1_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 ;<br/><b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> ;<br/><b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">b</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i>, <a class="ref" href="euclid.html#D8">EUCLID:def 8</a></i>;<br/><a NAME="E7:11_2_1_1_1_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p</font>,<font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E8:11_2_1_1_1_1"/><b>then </b><i><font color="Green">E88</font></i>: 
 <a href="topreal6.html#K1">dist</a> <font color="Maroon">x</font>,<font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="topreal6.html#D1">TOPREAL6:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_2_1_1_1_1_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="euclid.html#K21">`1</a> 
 ;<br/><a NAME="E2:11_2_1_1_1_1_1"/><i><font color="Green">E90</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/><a NAME="E3:11_2_1_1_1_1_1"/><i><font color="Green">E91</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/><a NAME="E4:11_2_1_1_1_1_1"/><b>then </b>
0 <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E5:11_2_1_1_1_1_1"/><b>then </b><i><font color="Green">E92</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K6">sqrt</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"><font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/><a NAME="E6:11_2_1_1_1_1_1"/>
 <a href="square_1.html#K6">sqrt</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"><font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="gobrd14.html#T9">GOBRD14:9</a></i>;<br/><a NAME="E7:11_2_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K3">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i>, <a class="ref" href="square_1.html#T78">SQUARE_1:78</a></i>;<br/><a NAME="E8:11_2_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K3">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> 
 <b>by </b><i><a class="ref" href="complex1.html#T161">COMPLEX1:161</a></i>;<br/><a NAME="E9:11_2_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> 0
 <b>by </b><i>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/><b>hence </b><a NAME="E10:11_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/></div><b>end;</b></div><a NAME="E10:11_2_1_1_1_1"/><b>then </b>
not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="topreal1.html#D14">TOPREAL1:def 14</a></i>;<br/><b>hence </b><a NAME="E11:11_2_1_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">a</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="yellow_6.html#T10">YELLOW_6:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:11_2_1_1_1_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">x</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> <a href="euclid.html#K22">`2</a> 
 ;<br/><div class="add"><b>take </b><font color="Maroon">r</font> = <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K22">`2</a> </span>)</span>;<br/><b>thus </b><a NAME="E2:11_2_1_1_1_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/><b>assume </b><a NAME="E3:11_2_1_1_1_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 ;<br/><b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> ;<br/><b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">b</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i>, <a class="ref" href="euclid.html#D8">EUCLID:def 8</a></i>;<br/><a NAME="E6:11_2_1_1_1_2"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p</font>,<font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/><a NAME="E7:11_2_1_1_1_2"/><b>then </b><i><font color="Green">E95</font></i>: 
 <a href="topreal6.html#K1">dist</a> <font color="Maroon">x</font>,<font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="topreal6.html#D1">TOPREAL6:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_2_1_1_1_2_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">c</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="euclid.html#K22">`2</a> 
 ;<br/><a NAME="E2:11_2_1_1_1_2_1"/><i><font color="Green">E97</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/><a NAME="E3:11_2_1_1_1_2_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/><a NAME="E4:11_2_1_1_1_2_1"/><b>then </b><i><font color="Green">E98</font></i>: 
0 <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E5:11_2_1_1_1_2_1"/><b>then </b><i><font color="Green">E99</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K6">sqrt</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"><font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/><a NAME="E6:11_2_1_1_1_2_1"/>
 <a href="square_1.html#K6">sqrt</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"><font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="gobrd14.html#T9">GOBRD14:9</a></i>;<br/><a NAME="E7:11_2_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K3">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> 
 <b>by </b><i>, <a class="ref" href="square_1.html#T78">SQUARE_1:78</a></i>;<br/><a NAME="E8:11_2_1_1_1_2_1"/><b>then </b><i><font color="Green">E100</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> 
 <b>by </b><i>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/><a NAME="E9:11_2_1_1_1_2_1"/><i><font color="Green">E101</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E10:11_2_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K22">`2</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E11:11_2_1_1_1_2_1"/><b>then </b><i><font color="Green">E102</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> 
 <b>by </b><i>, <a class="ref" href="square_1.html#T77">SQUARE_1:77</a></i>;<br/><a NAME="E12:11_2_1_1_1_2_1"/>
0 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E13:11_2_1_1_1_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><a NAME="E9:11_2_1_1_1_2"/><b>then </b>
not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="topreal1.html#D14">TOPREAL1:def 14</a></i>;<br/><b>hence </b><a NAME="E10:11_2_1_1_1_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">a</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="yellow_6.html#T10">YELLOW_6:10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:11_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">a</font></span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T22">TOPMETR:22</a></i>;<br/>
<b>hence </b><a NAME="E3:11_2_1"/>
 <a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">a</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>


</div>
