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

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><a NAME="E1:174"/><i><font color="Green">E80</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:174"/><i><font color="Green">E81</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E3:174"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="174_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:174_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/>

<a NAME="E2:174_1"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 
<b>by </b><i><a class="txt" href="xreal_1.html#E141"><i><font color="Green">Lemma70</font></i></a></i>;<br/>
<a NAME="E3:174_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> 2
 
<b>by </b><i><a class="txt" href="xreal_1.html#E1:174"><i><font color="Green">E80</font></i></a></i>;<br/>
<a NAME="E4:174_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 
<b>by </b><i><a class="txt" href="xreal_1.html#E1:174"><i><font color="Green">E80</font></i></a>, <a class="txt" href="xreal_1.html#E2:174_1"><i><font color="Green">E83</font></i></a></i>;<br/>
<a NAME="E5:174_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="xreal_1.html#E2:174"><i><font color="Green">E81</font></i></a>, <a class="txt" href="xreal_1.html#E107"><i><font color="Green">Lemma49</font></i></a></i>;<br/>
<a NAME="E6:174_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="xreal_1.html#E2:174_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="xreal_1.html#E3:174_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/>
<a NAME="E7:174_1"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="xreal_1.html#E1:174"><i><font color="Green">E80</font></i></a></i>;<br/>
<a NAME="E8:174_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="xreal_1.html#E2:174"><i><font color="Green">E81</font></i></a>, <a class="txt" href="xreal_1.html#E21"><i><font color="Green">Lemma16</font></i></a>, <a class="txt" href="xreal_1.html#E6:174_1"><i><font color="Green">E85</font></i></a></i>;<br/>
<a NAME="E9:174_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="xreal_1.html#E2:174_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="xreal_1.html#E3:174_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<b>hence </b><a NAME="E10:174_1"/>
contradiction
 <b>by </b><i><a class="txt" href="xreal_1.html#E2:174"><i><font color="Green">E81</font></i></a>, <a class="txt" href="xreal_1.html#E7:174_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="xreal_1.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="174_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:174_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 )
 </span><b>by </b><i><a class="txt" href="xreal_1.html#E3:174"><i><font color="Green">E82</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="174_2_1"><b>suppose </b></a><a NAME="E1:174_2_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:174_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E2:174"><i><font color="Green">E81</font></i></a>, <a class="txt" href="xreal_1.html#E72"><i><font color="Green">Lemma42</font></i></a></i>;<br/><a NAME="E3:174_2_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:174_2_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E5:174_2_1"/><i><font color="Green">E84</font></i>: 
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E6:174_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:174_2_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="xreal_1.html#E5:174_2_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="xreal_1.html#E21"><i><font color="Green">Lemma16</font></i></a></i>;<br/><a NAME="E7:174_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:174_2_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><b>hence </b><a NAME="E8:174_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:174"><i><font color="Green">E80</font></i></a>, <a class="txt" href="xreal_1.html#E5:174_2_1"><i><font color="Green">E84</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="174_2_2"><b>suppose </b></a><a NAME="E1:174_2_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:174_2_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:174"><i><font color="Green">E80</font></i></a></i>;<br/><b>hence </b><a NAME="E3:174_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="xreal_1.html#E2:174"><i><font color="Green">E81</font></i></a>, <a class="txt" href="xreal_1.html#E1:174_2_2"><i><font color="Green">E83</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
