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

<b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:111"/><i><font color="Green">E69</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E2:111"/><i><font color="Green">E70</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E3:111"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/> <a href="topreal6.html#K1">dist</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="topreal6.html#K1">dist</a> <font color="Olive">x</font>,<font color="Olive">y</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="jordan24.html#D1">JORDAN24:def 1</a><br/>

<b>let </b><font color="Maroon">p</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="E4:111"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">p</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/>
<i><font color="Green">E79</font></i>:  <a href="topreal6.html#K1">dist</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">p</font> = 
 <a href="square_1.html#K7">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"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">0 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan.html#E8:111"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="topreal6.html#T101">TOPREAL6:101</a></i>
<br/>.= 
 <a href="square_1.html#K7">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"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>

;<br/>
<div><i><font color="Green">E80</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="111_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:111_2"/><i><font color="Green">E81</font></i>: 
9 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> 
 ;<br/><a NAME="E2:111_2"/>
0 <a href="real_1.html#K3">+</a> 9 <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"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jordan.html#E39"><i><font color="Green">Lemma137</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E3:111_2"/><b>then </b>
3 <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K7">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"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan.html#T26" target="_self">Th26</a>, <a class="ref" href="square_1.html#T95">SQUARE_1:95</a></i>;<br/><a NAME="E4:111_2"/><b>then </b>
2 <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K7">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"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:111_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jordan.html#T29" target="_self">Th29</a>, , , <a class="ref" href="jordan.html#T27" target="_self">Th27</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E82</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="111_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:111_3"/><i><font color="Green">E83</font></i>: 
 <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> 
 ;<br/><a NAME="E2:111_3"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan6.html#K6">Vertical_Line</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan.html#E1:111_3"><i><font color="Green">E83</font></i></a>, </i>;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:111_3"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E5:111_3"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K6">Vertical_Line</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="jordan.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E7:111_3"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">x</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 <b>by </b><i><a class="ref" href="jordan.html#T33" target="_self">Th33</a>, <a class="ref" href="jordan6.html#T34">JORDAN6:34</a></i>;<br/><a NAME="E8:111_3"/><i><font color="Green">E117</font></i>: 
 <a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">x</font>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan.html#T32" target="_self">Th32</a>, <a class="ref" href="jordan1k.html#T29">JORDAN1K:29</a></i>;<br/><i><font color="Green">E118</font></i>:  <a href="topreal6.html#K1">dist</a> <font color="Maroon">x</font>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> = 
 <a href="square_1.html#K7">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="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</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="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="topreal6.html#T101">TOPREAL6:101</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 2</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</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="real_1.html#K5">-</a> 0</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan.html#T35" target="_self">Th35</a>, <a class="txt" href="jordan.html#E1:111_3"><i><font color="Green">E83</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">4 <a href="real_1.html#K3">+</a> <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="square_1.html#K5">^2</a> </span>)</span></span>)</span>

;<br/><div><i><font color="Green">E119</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="111_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:111_3_2"/>
 <a href="topreal6.html#K1">dist</a> <font color="Maroon">x</font>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="topreal6.html#K1">dist</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>
 ;<br/><a NAME="E2:111_3_2"/><b>then </b>
4 <a href="real_1.html#K3">+</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="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 4 <a href="real_1.html#K3">+</a> 0
 <b>by </b><i><a class="ref" href="jordan.html#T39" target="_self">Th39</a>, <a class="ref" href="jordan.html#T27" target="_self">Th27</a>, <a class="ref" href="square_1.html#T85">SQUARE_1:85</a>, <a class="ref" href="square_1.html#T94">SQUARE_1:94</a></i>;<br/><b>hence </b><a NAME="E3:111_3_2"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E11:111_3"/>
 <a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="jordan.html#T31" target="_self">Th31</a>, <a class="ref" href="jordan.html#T35" target="_self">Th35</a>, <a class="ref" href="jordan1k.html#T22">JORDAN1K:22</a></i>;<br/><a NAME="E12:111_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="topreal6.html#K1">dist</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="real_1.html#K3">+</a> 0
 <b>by </b><i><a class="ref" href="jordan.html#T37" target="_self">Th37</a>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E13:111_3"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E120</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="111_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:111_4"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><a NAME="E2:111_4"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan6.html#K6">Vertical_Line</a> 1
 <b>by </b><i><a class="txt" href="jordan.html#E8:111"><i><font color="Green">E82</font></i></a>, </i>;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:111_4"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E5:111_4"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K6">Vertical_Line</a> 1
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="jordan.html#T33" target="_self">Th33</a></i>;<br/><a NAME="E7:111_4"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">x</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="jordan.html#T35" target="_self">Th35</a>, <a class="ref" href="jordan6.html#T34">JORDAN6:34</a></i>;<br/><a NAME="E8:111_4"/><i><font color="Green">E178</font></i>: 
 <a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">x</font>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan.html#T33" target="_self">Th33</a>, <a class="ref" href="jordan1k.html#T29">JORDAN1K:29</a></i>;<br/><i><font color="Green">E180</font></i>:  <a href="topreal6.html#K1">dist</a> <font color="Maroon">x</font>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> = 
 <a href="square_1.html#K7">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="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23">|[</a><span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</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="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid.html#K23">|[</a><span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="topreal6.html#T101">TOPREAL6:101</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">4 <a href="real_1.html#K3">+</a> <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="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan.html#T37" target="_self">Th37</a>, <a class="txt" href="jordan.html#E8:111"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan.html#E13"><i><font color="Green">Lemma84</font></i></a></i>
;<br/><div><i><font color="Green">E181</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="111_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:111_4_2"/>
 <a href="topreal6.html#K1">dist</a> <font color="Maroon">x</font>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="topreal6.html#K1">dist</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>
 ;<br/><a NAME="E2:111_4_2"/><b>then </b>
4 <a href="real_1.html#K3">+</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="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 4 <a href="real_1.html#K3">+</a> 0
 <b>by </b><i>, <a class="ref" href="jordan.html#T27" target="_self">Th27</a>, <a class="ref" href="square_1.html#T85">SQUARE_1:85</a>, <a class="ref" href="square_1.html#T94">SQUARE_1:94</a></i>;<br/><b>hence </b><a NAME="E3:111_4_2"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E11:111_4"/>
 <a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="jordan.html#T32" target="_self">Th32</a>, <a class="ref" href="jordan.html#T37" target="_self">Th37</a>, <a class="ref" href="jordan1k.html#T22">JORDAN1K:22</a></i>;<br/><a NAME="E12:111_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="topreal6.html#K1">dist</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="real_1.html#K3">+</a> 0
 <b>by </b><i><a class="ref" href="jordan.html#T39" target="_self">Th39</a>, <a class="ref" href="jordan.html#T40" target="_self">Th40</a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E13:111_4"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan.html#T29" target="_self">Th29</a>, , </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E182</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="111_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:111_5"/>
 <a href="real_1.html#K1">-</a> 3 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> 
 ;<br/><a NAME="E2:111_5"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span> <a href="square_1.html#K5">^2</a> 
 <b>by </b><i><a class="ref" href="jgraph_5.html#T2">JGRAPH_5:2</a></i>;<br/><b>hence </b><a NAME="E3:111_5"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan.html#T30" target="_self">Th30</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:111"/>
3 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="ref" href="jordan.html#T30" target="_self">Th30</a>, , <a class="ref" href="square_1.html#T78">SQUARE_1:78</a></i>;<br/>
<b>hence </b><a NAME="E12:111"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#K2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span>,3
 <b>by </b><i><a class="txt" href="jordan.html#E39"><i><font color="Green">Lemma137</font></i></a>, <a class="ref" href="jordan.html#T31" target="_self">Th31</a>, <a class="ref" href="jordan.html#T32" target="_self">Th32</a></i>;<br/>


</div>
