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<b>let </b><font color="Maroon" title="c2">c</font>, <font color="Maroon" title="c3">b</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><a NAME="E1:174"/><i><font color="Green" title="E79">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 <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 holds <br/><font color="Maroon" title="c3">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">c</font>
 ;<br/>

<b>assume </b><a NAME="E2:174"/><i><font color="Green" title="E80">A2</font></i>: 
not <font color="Maroon" title="c2">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font>
 ;<br/>

<a NAME="E3:174"/><i><font color="Green" title="E81">A3</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<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" title="c3">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<a NAME="E2:174_1"/><b>then </b><i><font color="Green" title="E82">A4</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="xreal_1.html#E140"><i><font color="Green" title="E69">Lm45</font></i></a></i>;<br/>
<a NAME="E3:174_1"/><i><font color="Green" title="E83">A5</font></i>: 
<font color="Maroon" title="c2">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2
 
<b>by </b><i><a class="txt" href="xreal_1.html#E1:174"><i><font color="Green" title="E79">A1</font></i></a></i>;<br/>
<a NAME="E4:174_1"/>
<font color="Maroon" title="c2">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="xreal_1.html#E1:174"><i><font color="Green" title="E79">A1</font></i></a>, <a class="txt" href="xreal_1.html#E2:174_1"><i><font color="Green" title="E82">A4</font></i></a></i>;<br/>
<a NAME="E5:174_1"/><b>then </b>
<font color="Maroon" title="c3">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">c</font>
 
<b>by </b><i><a class="txt" href="xreal_1.html#E2:174"><i><font color="Green" title="E80">A2</font></i></a>, <a class="txt" href="xreal_1.html#E106"><i><font color="Green" title="E48">Lm32</font></i></a></i>;<br/>
<a NAME="E6:174_1"/><b>then </b><i><font color="Green" title="E84">A6</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="xreal_1.html#E2:174_1"><i><font color="Green" title="E82">A4</font></i></a>, <a class="txt" href="xreal_1.html#E3:174_1"><i><font color="Green" title="E83">A5</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>;<br/>
<a NAME="E7:174_1"/><b>then </b><i><font color="Green" title="E85">A7</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">c</font>
 
<b>by </b><i><a class="txt" href="xreal_1.html#E1:174"><i><font color="Green" title="E79">A1</font></i></a></i>;<br/>
<a NAME="E8:174_1"/>
<font color="Maroon" title="c3">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">c</font></span>)</span>
 
<b>by </b><i><a class="txt" href="xreal_1.html#E2:174"><i><font color="Green" title="E80">A2</font></i></a>, <a class="txt" href="xreal_1.html#E6:174_1"><i><font color="Green" title="E84">A6</font></i></a>, <a class="txt" href="xreal_1.html#E20"><i><font color="Green" title="E15">Lm13</font></i></a></i>;<br/>
<a NAME="E9:174_1"/><b>then </b>
<font color="Maroon" title="c3">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="xreal_1.html#E2:174_1"><i><font color="Green" title="E82">A4</font></i></a>, <a class="txt" href="xreal_1.html#E3:174_1"><i><font color="Green" title="E83">A5</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">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" title="E80">A2</font></i></a>, <a class="txt" href="xreal_1.html#E7:174_1"><i><font color="Green" title="E85">A7</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>
<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" title="c3">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><b>by </b><i><a class="txt" href="xreal_1.html#E3:174"><i><font color="Green" title="E81">A3</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" title="E82">A8</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:174_2_1"/><b>then </b>
<font color="Maroon" title="c3">b</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E2:174"><i><font color="Green" title="E80">A2</font></i></a>, <a class="txt" href="xreal_1.html#E71"><i><font color="Green" title="E41">Lm28</font></i></a></i>;<br/><a NAME="E3:174_2_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:174_2_1"><i><font color="Green" title="E82">A8</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>;<br/><b>then consider </b><font color="Maroon" title="c4">d</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> <b> such that </b><br/><a NAME="E5:174_2_1"/><i><font color="Green" title="E83">A9</font></i>: 
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c4">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T7" target="_self" title="XREAL_1:th.7">Th7</a></i>;<br/><a NAME="E6:174_2_1"/>
<font color="Maroon" title="c3">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:174_2_1"><i><font color="Green" title="E82">A8</font></i></a>, <a class="txt" href="xreal_1.html#E5:174_2_1"><i><font color="Green" title="E83">A9</font></i></a>, <a class="txt" href="xreal_1.html#E20"><i><font color="Green" title="E15">Lm13</font></i></a></i>;<br/><a NAME="E7:174_2_1"/><b>then </b>
<font color="Maroon" title="c3">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">c</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:174_2_1"><i><font color="Green" title="E82">A8</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">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" title="E79">A1</font></i></a>, <a class="txt" href="xreal_1.html#E5:174_2_1"><i><font color="Green" title="E83">A9</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" title="E82">A10</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:174_2_2"/>
<font color="Maroon" title="c3">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">c</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:174"><i><font color="Green" title="E79">A1</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" title="E80">A2</font></i></a>, <a class="txt" href="xreal_1.html#E1:174_2_2"><i><font color="Green" title="E82">A10</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
