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<div class="add">

<b>set </b><font color="Maroon" title="c2">X</font> =  <a href="topreal1.html#K7" title="TOPREAL1:func.7">east_halfline</a> <font color="Maroon" title="c1">a</font>;<br/>
<a NAME="E1:11_3_1"/>
for <font color="Olive" title="b1">p</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#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K7" title="TOPREAL1:func.7">east_halfline</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  holds <br/> ex <font color="Olive" title="b2">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K7" title="TOPREAL1:func.7">east_halfline</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">p</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#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:11_3_1_1"/><i><font color="Green" title="E9">A39</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K7" title="TOPREAL1:func.7">east_halfline</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c3">p</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> 2</span>)</span> ;<br/>
<a NAME="E3:11_3_1_1"/><i><font color="Green" title="E10">A40</font></i>: 
not <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K7" title="TOPREAL1:func.7">east_halfline</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="txt" href="jordan18.html#E1:11_3_1_1"><i><font color="Green" title="E9">A39</font></i></a>, <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="11_3_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_3_1_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  or <font color="Maroon" title="c4">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 </span><b>by </b><i><a class="txt" href="jordan18.html#E3:11_3_1_1"><i><font color="Green" title="E10">A40</font></i></a>, <a class="ref" href="topreal1.html#D13" title="TOPREAL1:def.13">TOPREAL1:def 13</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:11_3_1_1_1_1"/><i><font color="Green" title="E11">A41</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c5">r</font> =  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span>;<br/><a NAME="E2:11_3_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="jordan18.html#E1:11_3_1_1_1_1"><i><font color="Green" title="E11">A41</font></i></a></i>;<br/><b>hence </b><a NAME="E3:11_3_1_1_1_1"/>
<font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T133" title="COMPLEX1:th.133">COMPLEX1:133</a></i>;<br/><b>let </b><font color="Maroon" title="c6">b</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="E4:11_3_1_1_1_1"/><i><font color="Green" title="E12">A42</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">r</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c6">b</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#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> ;<br/><b>reconsider </b><font color="Maroon" title="c8">c</font> = <font color="Maroon" title="c7">b</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> 2</span>)</span> ;<br/><a NAME="E7:11_3_1_1_1_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c7">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="txt" href="jordan18.html#E4:11_3_1_1_1_1"><i><font color="Green" title="E12">A42</font></i></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/><a NAME="E8:11_3_1_1_1_1"/><b>then </b><i><font color="Green" title="E13">A43</font></i>: 
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c8">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="ref" href="topreal6.html#D1" title="TOPREAL6:def.1">TOPREAL6:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_3_1_1_1_1_1"/><i><font color="Green" title="E14">A44</font></i>: 
<font color="Maroon" title="c8">c</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 ;<br/><a NAME="E2:11_3_1_1_1_1_1"/><i><font color="Green" title="E15">A45</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></i>;<br/><a NAME="E3:11_3_1_1_1_1_1"/><i><font color="Green" title="E16">A46</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></i>;<br/><a NAME="E4:11_3_1_1_1_1_1"/><b>then </b>
0 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jordan18.html#E2:11_3_1_1_1_1_1"><i><font color="Green" title="E15">A45</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E5:11_3_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E17">A47</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K6" title="SQUARE_1:func.6">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" title="c4">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/><a NAME="E6:11_3_1_1_1_1_1"/>
 <a href="square_1.html#K6" title="SQUARE_1:func.6">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" title="c4">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan18.html#E8:11_3_1_1_1_1"><i><font color="Green" title="E13">A43</font></i></a>, <a class="txt" href="jordan18.html#E1:11_3_1_1_1_1_1"><i><font color="Green" title="E14">A44</font></i></a>, <a class="ref" href="topreal6.html#T101" title="TOPREAL6:th.101">TOPREAL6:101</a></i>;<br/><a NAME="E7:11_3_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">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" title="c4">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 <b>by </b><i><a class="txt" href="jordan18.html#E5:11_3_1_1_1_1_1"><i><font color="Green" title="E17">A47</font></i></a>, <a class="ref" href="square_1.html#T78" title="SQUARE_1:th.78">SQUARE_1:78</a></i>;<br/><a NAME="E8:11_3_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">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" title="c4">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> 
 <b>by </b><i><a class="ref" href="complex1.html#T161" title="COMPLEX1:th.161">COMPLEX1:161</a></i>;<br/><a NAME="E9:11_3_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" title="c4">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0
 <b>by </b><i><a class="txt" href="jordan18.html#E3:11_3_1_1_1_1_1"><i><font color="Green" title="E16">A46</font></i></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/><b>hence </b><a NAME="E10:11_3_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan18.html#E2:11_3_1_1_1_1_1"><i><font color="Green" title="E15">A45</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><a NAME="E10:11_3_1_1_1_1"/><b>then </b>
not <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K7" title="TOPREAL1:func.7">east_halfline</a> <font color="Maroon" title="c1">a</font>
 <b>by </b><i><a class="ref" href="topreal1.html#D13" title="TOPREAL1:def.13">TOPREAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E11:11_3_1_1_1_1"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K7" title="TOPREAL1:func.7">east_halfline</a> <font color="Maroon" title="c1">a</font></span>)</span> <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:11_3_1_1_1_2"/><i><font color="Green" title="E11">A48</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c5">r</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>;<br/><b>thus </b><a NAME="E2:11_3_1_1_1_2"/>
<font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jordan18.html#E1:11_3_1_1_1_2"><i><font color="Green" title="E11">A48</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/><b>let </b><font color="Maroon" title="c6">b</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="E3:11_3_1_1_1_2"/><i><font color="Green" title="E12">A49</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">r</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c6">b</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#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> ;<br/><b>reconsider </b><font color="Maroon" title="c8">c</font> = <font color="Maroon" title="c7">b</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> 2</span>)</span> ;<br/><a NAME="E6:11_3_1_1_1_2"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c7">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="txt" href="jordan18.html#E3:11_3_1_1_1_2"><i><font color="Green" title="E12">A49</font></i></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/><a NAME="E7:11_3_1_1_1_2"/><b>then </b><i><font color="Green" title="E13">A50</font></i>: 
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c8">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="ref" href="topreal6.html#D1" title="TOPREAL6:def.1">TOPREAL6:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_3_1_1_1_2_1"/><i><font color="Green" title="E14">A51</font></i>: 
<font color="Maroon" title="c8">c</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 ;<br/><a NAME="E2:11_3_1_1_1_2_1"/><i><font color="Green" title="E15">A52</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></i>;<br/><a NAME="E3:11_3_1_1_1_2_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></i>;<br/><a NAME="E4:11_3_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E16">A53</font></i>: 
0 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jordan18.html#E2:11_3_1_1_1_2_1"><i><font color="Green" title="E15">A52</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E5:11_3_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E17">A54</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K6" title="SQUARE_1:func.6">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" title="c4">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/><a NAME="E6:11_3_1_1_1_2_1"/>
 <a href="square_1.html#K6" title="SQUARE_1:func.6">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" title="c4">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jordan18.html#E7:11_3_1_1_1_2"><i><font color="Green" title="E13">A50</font></i></a>, <a class="ref" href="topreal6.html#T101" title="TOPREAL6:th.101">TOPREAL6:101</a></i>;<br/><a NAME="E7:11_3_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">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" title="c4">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> 
 <b>by </b><i><a class="txt" href="jordan18.html#E5:11_3_1_1_1_2_1"><i><font color="Green" title="E17">A54</font></i></a>, <a class="ref" href="square_1.html#T78" title="SQUARE_1:th.78">SQUARE_1:78</a></i>;<br/><a NAME="E8:11_3_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E18">A55</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> 
 <b>by </b><i><a class="txt" href="jordan18.html#E4:11_3_1_1_1_2_1"><i><font color="Green" title="E16">A53</font></i></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/><a NAME="E9:11_3_1_1_1_2_1"/><i><font color="Green" title="E19">A56</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jordan18.html#E1:11_3_1_1_1_2"><i><font color="Green" title="E11">A48</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/><a NAME="E10:11_3_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jordan18.html#E1:11_3_1_1_1_2_1"><i><font color="Green" title="E14">A51</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E11:11_3_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E20">A57</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> 
 <b>by </b><i><a class="txt" href="jordan18.html#E9:11_3_1_1_1_2_1"><i><font color="Green" title="E19">A56</font></i></a>, <a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></i>;<br/><a NAME="E12:11_3_1_1_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jordan18.html#E2:11_3_1_1_1_2_1"><i><font color="Green" title="E15">A52</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E13:11_3_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan18.html#E8:11_3_1_1_1_2_1"><i><font color="Green" title="E18">A55</font></i></a>, <a class="txt" href="jordan18.html#E11:11_3_1_1_1_2_1"><i><font color="Green" title="E20">A57</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="E9:11_3_1_1_1_2"/><b>then </b>
not <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K7" title="TOPREAL1:func.7">east_halfline</a> <font color="Maroon" title="c1">a</font>
 <b>by </b><i><a class="ref" href="topreal1.html#D13" title="TOPREAL1:def.13">TOPREAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E10:11_3_1_1_1_2"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K7" title="TOPREAL1:func.7">east_halfline</a> <font color="Maroon" title="c1">a</font></span>)</span> <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></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:11_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K7" title="TOPREAL1:func.7">east_halfline</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a></i>;<br/>
<b>hence </b><a NAME="E3:11_3_1"/>
 <a href="topreal1.html#K7" title="TOPREAL1:func.7">east_halfline</a> <font color="Maroon" title="c1">a</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></i>;<br/>


</div>
