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

<b>let </b><font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:179"/><i><font color="Green" title="E83">A1</font></i>: 
for <font color="Olive" title="b1">a</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 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 holds <br/><font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">c</font>
 <b>and </b><br/><a NAME="E2:179"/><i><font color="Green" title="E84">A2</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>

<a NAME="E3:179"/><i><font color="Green" title="E85">A3</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="txt" href="xreal_1.html#E2:179"><i><font color="Green" title="E84">A2</font></i></a>, <a class="ref" href="xreal_1.html#T157" target="_self" title="XREAL_1:th.157">Th157</a></i>;<br/>
<a NAME="E4:179"/><b>then </b><i><font color="Green" title="E86">A4</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T85" target="_self" title="XREAL_1:th.85">Th85</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:179_1"/>
( <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 or ( <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 ) or ( <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_1_1"><b>suppose </b></a><a NAME="E1:179_1_1"/>
<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:179_1_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T133" target="_self" title="XREAL_1:th.133">Th133</a></i>;<br/><b>hence </b><a NAME="E3:179_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:179"><i><font color="Green" title="E83">A1</font></i></a>, <a class="txt" href="xreal_1.html#E2:179"><i><font color="Green" title="E84">A2</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:179_1_2"/><i><font color="Green" title="E87">A5</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font>
 <b>and </b><br/><a NAME="E2:179_1_2"/><i><font color="Green" title="E88">A6</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c4">a</font> = <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span>;<br/><a NAME="E3:179_1_2"/>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="xreal_1.html#E2:179"><i><font color="Green" title="E84">A2</font></i></a>, <a class="txt" href="xreal_1.html#E138"><i><font color="Green" title="E65">Lm43</font></i></a></i>;<br/><a NAME="E4:179_1_2"/><b>then </b><i><font color="Green" title="E89">A7</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="xreal_1.html#E2:179_1_2"><i><font color="Green" title="E88">A6</font></i></a>, <a class="txt" href="xreal_1.html#E131"><i><font color="Green" title="E63">Lm41</font></i></a></i>;<br/><a NAME="E5:179_1_2"/><i><font color="Green" title="E90">A8</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="xreal_1.html#E2:179_1_2"><i><font color="Green" title="E88">A6</font></i></a>, <a class="txt" href="xreal_1.html#E1:179_1_2"><i><font color="Green" title="E87">A5</font></i></a>, <a class="txt" href="xreal_1.html#E221"><i><font color="Green" title="E81">Lm50</font></i></a></i>;<br/><a NAME="E6:179_1_2"/>
<font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="xreal_1.html#E2:179"><i><font color="Green" title="E84">A2</font></i></a>, <a class="txt" href="xreal_1.html#E2:179_1_2"><i><font color="Green" title="E88">A6</font></i></a>, <a class="txt" href="xreal_1.html#E131"><i><font color="Green" title="E63">Lm41</font></i></a></i>;<br/><a NAME="E7:179_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T157" target="_self" title="XREAL_1:th.157">Th157</a></i>;<br/><a NAME="E8:179_1_2"/><b>then </b>
<font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T85" target="_self" title="XREAL_1:th.85">Th85</a></i>;<br/><a NAME="E9:179_1_2"/><b>then </b>
<font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T79" title="XCMPLX_1:th.79">XCMPLX_1:79</a></i>;<br/><a NAME="E10:179_1_2"/><b>then </b>
<font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <b>by </b><i><a class="txt" href="xreal_1.html#E5:179_1_2"><i><font color="Green" title="E90">A8</font></i></a>, <a class="txt" href="xreal_1.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a></i>;<br/><a NAME="E11:179_1_2"/><b>then </b><i><font color="Green" title="E91">A9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:179"><i><font color="Green" title="E83">A1</font></i></a>, <a class="txt" href="xreal_1.html#E4:179_1_2"><i><font color="Green" title="E89">A7</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:179_1_2_1"/>
( <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">b</font> or <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_1_2_1_1"><b>suppose </b></a><a NAME="E1:179_1_2_1_1"/>
<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">b</font>
 ;<br/><div class="add"><a NAME="E2:179_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:179_1_2"><i><font color="Green" title="E87">A5</font></i></a>, <a class="txt" href="xreal_1.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a></i>;<br/><a NAME="E3:179_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <b>by </b><i><a class="txt" href="xreal_1.html#E2:179_1_2"><i><font color="Green" title="E88">A6</font></i></a>, <a class="ref" href="xcmplx_1.html#T93" title="XCMPLX_1:th.93">XCMPLX_1:93</a></i>;<br/><b>hence </b><a NAME="E4:179_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="xreal_1.html#E3:179"><i><font color="Green" title="E85">A3</font></i></a>, <a class="txt" href="xreal_1.html#E11:179_1_2"><i><font color="Green" title="E91">A9</font></i></a>, <a class="ref" href="xreal_1.html#T85" target="_self" title="XREAL_1:th.85">Th85</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_1_2_1_2"><b>suppose </b></a><a NAME="E1:179_1_2_1_2"/><i><font color="Green" title="E92">A10</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font>
 ;<br/><div class="add"><a NAME="E2:179_1_2_1_2"/>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="xreal_1.html#E2:179"><i><font color="Green" title="E84">A2</font></i></a>, <a class="txt" href="xreal_1.html#E138"><i><font color="Green" title="E65">Lm43</font></i></a></i>;<br/><a NAME="E3:179_1_2_1_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="xreal_1.html#E2:179_1_2"><i><font color="Green" title="E88">A6</font></i></a>, <a class="txt" href="xreal_1.html#E131"><i><font color="Green" title="E63">Lm41</font></i></a></i>;<br/><a NAME="E4:179_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E2:179_1_2"><i><font color="Green" title="E88">A6</font></i></a>, <a class="txt" href="xreal_1.html#E1:179_1_2_1_2"><i><font color="Green" title="E92">A10</font></i></a>, <a class="ref" href="xreal_1.html#T100" target="_self" title="XREAL_1:th.100">Th100</a></i>;<br/><a NAME="E5:179_1_2_1_2"/><b>then </b><i><font color="Green" title="E93">A11</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <b>by </b><i><a class="txt" href="xreal_1.html#E2:179"><i><font color="Green" title="E84">A2</font></i></a>, <a class="ref" href="xcmplx_1.html#T93" title="XCMPLX_1:th.93">XCMPLX_1:93</a></i>;<br/><a NAME="E6:179_1_2_1_2"/>
<font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:179_1_2_1_2"><i><font color="Green" title="E92">A10</font></i></a>, <a class="txt" href="xreal_1.html#E74"><i><font color="Green" title="E42">Lm31</font></i></a></i>;<br/><a NAME="E7:179_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <b>by </b><i><a class="txt" href="xreal_1.html#E5:179_1_2_1_2"><i><font color="Green" title="E93">A11</font></i></a>, <a class="txt" href="xreal_1.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a></i>;<br/><b>hence </b><a NAME="E8:179_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="xreal_1.html#E11:179_1_2"><i><font color="Green" title="E91">A9</font></i></a>, <a class="txt" href="xreal_1.html#E4:179"><i><font color="Green" title="E86">A4</font></i></a>, <a class="txt" href="xreal_1.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:179_1_3"/><i><font color="Green" title="E87">A12</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">c</font>
 <b>and </b><br/><a NAME="E2:179_1_3"/><i><font color="Green" title="E88">A13</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c4">a</font> = <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">c</font></span>)</span>;<br/><a NAME="E3:179_1_3"/>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="xreal_1.html#E2:179_1_3"><i><font color="Green" title="E88">A13</font></i></a>, <a class="txt" href="xreal_1.html#E138"><i><font color="Green" title="E65">Lm43</font></i></a></i>;<br/><a NAME="E4:179_1_3"/><b>then </b><i><font color="Green" title="E89">A14</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="xreal_1.html#E2:179"><i><font color="Green" title="E84">A2</font></i></a>, <a class="txt" href="xreal_1.html#E131"><i><font color="Green" title="E63">Lm41</font></i></a></i>;<br/><a NAME="E5:179_1_3"/><i><font color="Green" title="E90">A15</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="xreal_1.html#E2:179_1_3"><i><font color="Green" title="E88">A13</font></i></a>, <a class="txt" href="xreal_1.html#E1:179_1_3"><i><font color="Green" title="E87">A12</font></i></a>, <a class="txt" href="xreal_1.html#E222"><i><font color="Green" title="E82">Lm51</font></i></a></i>;<br/><a NAME="E6:179_1_3"/>
<font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="xreal_1.html#E2:179"><i><font color="Green" title="E84">A2</font></i></a>, <a class="txt" href="xreal_1.html#E2:179_1_3"><i><font color="Green" title="E88">A13</font></i></a>, <a class="txt" href="xreal_1.html#E131"><i><font color="Green" title="E63">Lm41</font></i></a></i>;<br/><a NAME="E7:179_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T157" target="_self" title="XREAL_1:th.157">Th157</a></i>;<br/><a NAME="E8:179_1_3"/><b>then </b>
<font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T85" target="_self" title="XREAL_1:th.85">Th85</a></i>;<br/><a NAME="E9:179_1_3"/><b>then </b>
<font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T79" title="XCMPLX_1:th.79">XCMPLX_1:79</a></i>;<br/><a NAME="E10:179_1_3"/><b>then </b>
<font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <b>by </b><i><a class="txt" href="xreal_1.html#E5:179_1_3"><i><font color="Green" title="E90">A15</font></i></a>, <a class="txt" href="xreal_1.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a></i>;<br/><a NAME="E11:179_1_3"/><b>then </b><i><font color="Green" title="E91">A16</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:179"><i><font color="Green" title="E83">A1</font></i></a>, <a class="txt" href="xreal_1.html#E4:179_1_3"><i><font color="Green" title="E89">A14</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:179_1_3_1"/>
( <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">b</font> or <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_1_3_1_1"><b>suppose </b></a><a NAME="E1:179_1_3_1_1"/>
<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">b</font>
 ;<br/><div class="add"><a NAME="E2:179_1_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <b>by </b><i><a class="txt" href="xreal_1.html#E2:179_1_3"><i><font color="Green" title="E88">A13</font></i></a>, <a class="ref" href="xcmplx_1.html#T93" title="XCMPLX_1:th.93">XCMPLX_1:93</a></i>;<br/><b>hence </b><a NAME="E3:179_1_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="xreal_1.html#E3:179"><i><font color="Green" title="E85">A3</font></i></a>, <a class="txt" href="xreal_1.html#E11:179_1_3"><i><font color="Green" title="E91">A16</font></i></a>, <a class="ref" href="xreal_1.html#T85" target="_self" title="XREAL_1:th.85">Th85</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_1_3_1_2"><b>suppose </b></a><a NAME="E1:179_1_3_1_2"/>
<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:179_1_3_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:179_1_3"><i><font color="Green" title="E87">A12</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
