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

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> ;<br/>
<a NAME="E2:56_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:56_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:56_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 ;<br/>

<a NAME="E2:56_1_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:56_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:56_1_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E4:56_1_1"/><i><font color="Green">E39</font></i>: 
ex <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="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_2"><b>proof </b></a><div class="add">

<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</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="E1:56_1_1_2_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>set </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K5">-</a> 1;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:56_1_1_2_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="seq_4.html#T10">SEQ_4:10</a></i>;<br/><a NAME="E5:56_1_1_2_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="power.html#E1:56_1_1_2_1"><i><font color="Green">E41</font></i></a></i>;<br/><a NAME="E6:56_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:56_1_1_2_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="power.html#K4" target="_self">to_power</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T56" target="_self">Th56</a></i>;<br/><a NAME="E8:56_1_1_2_1"/>
1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E9:56_1_1_2_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="power.html#K4" target="_self">to_power</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="power.html#E7:56_1_1_2_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:56_1_1_2_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="power.html#E1:56_1_1_2_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E11:56_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="power.html#E4:56_1_1_2_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E12:56_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="power.html#E10:56_1_1_2_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xcmplx_1.html#T93">XCMPLX_1:93</a></i>;<br/><a NAME="E13:56_1_1_2_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="power.html#E9:56_1_1_2_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E14:56_1_1_2_1"/>
1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="power.html#E1:56_1_1_2_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><a NAME="E15:56_1_1_2_1"/><b>then </b>
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="power.html#E13:56_1_1_2_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="real_2.html#T152">REAL_2:152</a></i>;<br/><a NAME="E16:56_1_1_2_1"/><b>then </b>
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T56">XCMPLX_1:56</a></i>;<br/><a NAME="E17:56_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="power.html#K4" target="_self">to_power</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="power.html#E5:56_1_1_2_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="power.html#T33" target="_self">Th33</a></i>;<br/><b>hence </b><a NAME="E18:56_1_1_2_1"/>
ex <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="Maroon">c<sub>7</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1_1_2_2_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 or <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 </span><b>by </b><i><a class="txt" href="power.html#E1:56"><i><font color="Green">E37</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:56_1_1_2_2_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:56_1_1_2_2_1_1"/>
ex <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="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="power.html#E2:56"><i><font color="Green">E38</font></i></a>, <a class="txt" href="power.html#E1:56_1_1_2"><i><font color="Green">E40</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:56_1_1_2_2_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><div class="add"><a NAME="E2:56_1_1_2_2_1_2"/><b>then </b>
1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K6">/</a> 1
 <b>by </b><i><a class="txt" href="power.html#E1:56"><i><font color="Green">E37</font></i></a>, <a class="ref" href="real_2.html#T151">REAL_2:151</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>7</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="E4:56_1_1_2_2_1_2"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="power.html#E2:56"><i><font color="Green">E38</font></i></a>, <a class="txt" href="power.html#E1:56_1_1_2"><i><font color="Green">E40</font></i></a></i>;<br/><a NAME="E5:56_1_1_2_2_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="power.html#K3" target="_self">to_power</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="power.html#E1:56"><i><font color="Green">E37</font></i></a>, <a class="txt" href="power.html#E4:56_1_1_2_2_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="power.html#T37" target="_self">Th37</a></i>;<br/><b>hence </b><a NAME="E6:56_1_1_2_2_1_2"/>
ex <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="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>7</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="E4:56_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<a NAME="E5:56_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:56_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="power.html#E4:56_1_1_2"><i><font color="Green">E41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1_1_3_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 or <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 </span><b>by </b><i><a class="txt" href="power.html#E1:56"><i><font color="Green">E37</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:56_1_1_3_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:56_1_1_3_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="seq_4.html#V1">bounded_above</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_3_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E2:56_1_1_3_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="seq_4.html#T10">SEQ_4:10</a></i>;<br/>
<a NAME="E3:56_1_1_3_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="txt" href="power.html#E1:56_1_1_3_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E4:56_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="txt" href="power.html#E2:56_1_1_3_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="real_2.html#T177">REAL_2:177</a></i>;<br/>
<a NAME="E5:56_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E6:56_1_1_3_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="txt" href="power.html#E1:56"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E7:56_1_1_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="power.html#K4" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="power.html#T56" target="_self">Th56</a></i>;<br/>
<a NAME="E8:56_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="power.html#E5:56_1_1_3_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_3_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</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="E1:56_1_1_3_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:56_1_1_3_1_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><a NAME="E3:56_1_1_3_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="power.html#K4" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="power.html#E8:56_1_1_3_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E4:56_1_1_3_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="power.html#E1:56_1_1_3_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="power.html#T44" target="_self">Th44</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:56_1_1_3_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="seq_4.html#V1">bounded_above</a>
 <b>by </b><i><a class="ref" href="seq_4.html#D1">SEQ_4:def 1</a></i>;<br/>


</div><b>end;</b></div><b>take </b><font color="Maroon">c<sub>7</sub></font> =  <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>6</sub></font>;<br/><a NAME="E3:56_1_1_3_1_1"/><i><font color="Green">E42</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_3_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:56_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</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="E3:56_1_1_3_1_1_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T7">XREAL_1:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:56_1_1_3_1_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="seq_4.html#T10">SEQ_4:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> =  <a href="xxreal_0.html#K4">max</a> 1,<font color="Maroon">c<sub>10</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="xxreal_0.html#T16">XXREAL_0:16</a></i>;<br/>
<a NAME="E8:56_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E9:56_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="power.html#E6:56_1_1_3_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E10:56_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>9</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="power.html#E2:56"><i><font color="Green">E38</font></i></a>, <a class="txt" href="power.html#E3:56_1_1_3_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="real_2.html#T142">REAL_2:142</a></i>;<br/>
<a NAME="E11:56_1_1_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E12:56_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="power.html#E9:56_1_1_3_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="real_2.html#T177">REAL_2:177</a></i>;<br/>
<a NAME="E13:56_1_1_3_1_1_2"/><i><font color="Green">E47</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E14:56_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font>
 
;<br/>
<a NAME="E15:56_1_1_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="txt" href="power.html#E12:56_1_1_3_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="real_2.html#T178">REAL_2:178</a></i>;<br/>
<a NAME="E16:56_1_1_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E17:56_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E18:56_1_1_3_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="power.html#K2" target="_self">-root</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="power.html#E1:56"><i><font color="Green">E37</font></i></a>, <a class="txt" href="power.html#E13:56_1_1_3_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="power.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E19:56_1_1_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="power.html#K1" target="_self">-root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E20:56_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="power.html#K1" target="_self">-root</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</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="power.html#E17:56_1_1_3_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E21:56_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="power.html#K4" target="_self">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="power.html#E1:56"><i><font color="Green">E37</font></i></a>, <a class="txt" href="power.html#E13:56_1_1_3_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="power.html#T52" target="_self">Th52</a></i>;<br/>
<a NAME="E22:56_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="power.html#E2:56"><i><font color="Green">E38</font></i></a>, <a class="ref" href="real_2.html#T178">REAL_2:178</a></i>;<br/>
<a NAME="E23:56_1_1_3_1_1_2"/>
1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="power.html#E14:56_1_1_3_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</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="E25:56_1_1_3_1_1_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="power.html#E4:56_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="power.html#E2:56_1_1_3_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E27:56_1_1_3_1_1_2"/><i><font color="Green">E52</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="power.html#E25:56_1_1_3_1_1_2"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E28:56_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="power.html#K4" target="_self">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="power.html#E1:56"><i><font color="Green">E37</font></i></a>, <a class="ref" href="power.html#T39" target="_self">Th39</a></i>;<br/>
<a NAME="E29:56_1_1_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="power.html#E27:56_1_1_3_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E30:56_1_1_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="power.html#E22:56_1_1_3_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E31:56_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="power.html#K4" target="_self">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="power.html#E1:56"><i><font color="Green">E37</font></i></a>, <a class="ref" href="power.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E32:56_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="power.html#E25:56_1_1_3_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E33:56_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="power.html#K4" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="power.html#K4" target="_self">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="power.html#E1:56_1_1_3_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="power.html#T44" target="_self">Th44</a></i>;<br/>
<b>hence </b><a NAME="E34:56_1_1_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="power.html#E3:56_1_1_3_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="power.html#E31:56_1_1_3_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:56_1_1_3_1_1"/>
not <font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_3_1_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:56_1_1_3_1_1_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</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="E3:56_1_1_3_1_1_3"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</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="ref" href="xreal_1.html#T7">XREAL_1:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E5:56_1_1_3_1_1_3"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="power.html#K4" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="power.html#E1:56"><i><font color="Green">E37</font></i></a>, <a class="ref" href="power.html#T39" target="_self">Th39</a></i>;<br/>
<a NAME="E6:56_1_1_3_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="power.html#E3:56_1_1_3_1_1_3"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E7:56_1_1_3_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="power.html#E3:56_1_1_3_1_1_3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="real_2.html#T142">REAL_2:142</a></i>;<br/>
<a NAME="E8:56_1_1_3_1_1_3"/><b>then </b><i><font color="Green">E45</font></i>: 
<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>9</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  = <font color="Maroon">a<sub>1</sub></font> <a href="power.html#K1" target="_self">-root</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E10:56_1_1_3_1_1_3"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<a NAME="E11:56_1_1_3_1_1_3"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 holds <br/><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="power.html#K2" target="_self">-root</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="power.html#E10:56_1_1_3_1_1_3"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E12:56_1_1_3_1_1_3"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="power.html#E1:56"><i><font color="Green">E37</font></i></a>, <a class="ref" href="power.html#T24" target="_self">Th24</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E14:56_1_1_3_1_1_3"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>11</sub></font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</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>9</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="power.html#E8:56_1_1_3_1_1_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> =  <a href="xxreal_0.html#K4">max</a> 1,<font color="Maroon">c<sub>11</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="xxreal_0.html#T16">XXREAL_0:16</a></i>;<br/>
<a NAME="E16:56_1_1_3_1_1_3"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E17:56_1_1_3_1_1_3"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</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>9</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="txt" href="power.html#E14:56_1_1_3_1_1_3"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E18:56_1_1_3_1_1_3"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="power.html#K1" target="_self">-root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</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>9</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="txt" href="power.html#E10:56_1_1_3_1_1_3"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E19:56_1_1_3_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="power.html#K1" target="_self">-root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="power.html#K1" target="_self">-root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/>
<a NAME="E20:56_1_1_3_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="power.html#K1" target="_self">-root</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</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>9</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="txt" href="power.html#E18:56_1_1_3_1_1_3"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E21:56_1_1_3_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="power.html#K1" target="_self">-root</a> <font color="Maroon">c<sub>1</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>9</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E22:56_1_1_3_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="power.html#K4" target="_self">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="power.html#E1:56"><i><font color="Green">E37</font></i></a>, <a class="txt" href="power.html#E16:56_1_1_3_1_1_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="power.html#T52" target="_self">Th52</a></i>;<br/>
<a NAME="E23:56_1_1_3_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</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>9</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="power.html#E5:56_1_1_3_1_1_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E24:56_1_1_3_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="power.html#K4" target="_self">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="power.html#K4" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="power.html#E1:56"><i><font color="Green">E37</font></i></a>, <a class="ref" href="power.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E25:56_1_1_3_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>;<br/>
<a NAME="E26:56_1_1_3_1_1_3"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>;<br/>
<a NAME="E27:56_1_1_3_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="power.html#E3:56_1_1_3_1_1_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="power.html#E5:56_1_1_3_1_1_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="real_2.html#T142">REAL_2:142</a></i>;<br/>
<a NAME="E28:56_1_1_3_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="power.html#E2:56"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E29:56_1_1_3_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <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="power.html#E26:56_1_1_3_1_1_3"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E30:56_1_1_3_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 
;<br/>
<a NAME="E31:56_1_1_3_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="power.html#E2:56_1_1_3_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<a NAME="E32:56_1_1_3_1_1_3"/><b>then </b><i><font color="Green">E51</font></i>: 
1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i><a class="ref" href="real_2.html#T174">REAL_2:174</a></i>;<br/>
<a NAME="E33:56_1_1_3_1_1_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="power.html#E16:56_1_1_3_1_1_3"><i><font color="Green">E48</font></i></a></i>;<br/>
<b>hence </b><a NAME="E34:56_1_1_3_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="power.html#E32:56_1_1_3_1_1_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:56_1_1_3_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="power.html#E3:56_1_1_3_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:56_1_1_3_1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><div class="add"><a NAME="E2:56_1_1_3_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="seq_4.html#V2">bounded_below</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_3_1_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E2:56_1_1_3_1_2_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="seq_4.html#T10">SEQ_4:10</a></i>;<br/>
<a NAME="E3:56_1_1_3_1_2_1"/>
1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K6">/</a> 1
 
<b>by </b><i><a class="txt" href="power.html#E1:56"><i><font color="Green">E37</font></i></a>, <a class="txt" href="power.html#E1:56_1_1_3_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="real_2.html#T151">REAL_2:151</a></i>;<br/>
<a NAME="E4:56_1_1_3_1_2_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E5:56_1_1_3_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="txt" href="power.html#E2:56_1_1_3_1_2_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="real_2.html#T177">REAL_2:177</a></i>;<br/>
<a NAME="E6:56_1_1_3_1_2_1"/><b>then </b><i><font color="Green">E43</font></i>: 
1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E7:56_1_1_3_1_2_1"/>
1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="power.html#E1:56"><i><font color="Green">E37</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E8:56_1_1_3_1_2_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E9:56_1_1_3_1_2_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="power.html#K4" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="power.html#T56" target="_self">Th56</a></i>;<br/>
<a NAME="E10:56_1_1_3_1_2_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="power.html#K4" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>
 
;<br/>
<a NAME="E11:56_1_1_3_1_2_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="power.html#E6:56_1_1_3_1_2_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:56_1_1_3_1_2_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="power.html#K4" target="_self">to_power</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="power.html#E1:56"><i><font color="Green">E37</font></i></a>, <a class="ref" href="power.html#T37" target="_self">Th37</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_3_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</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="E1:56_1_1_3_1_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:56_1_1_3_1_2_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><a NAME="E3:56_1_1_3_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="power.html#K4" target="_self">to_power</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="power.html#E12:56_1_1_3_1_2_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E4:56_1_1_3_1_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="power.html#E1:56"><i><font color="Green">E37</font></i></a>, <a class="txt" href="power.html#E1:56_1_1_3_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="power.html#T45" target="_self">Th45</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:56_1_1_3_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="seq_4.html#V2">bounded_below</a>
 <b>by </b><i><a class="ref" href="seq_4.html#D2">SEQ_4:def 2</a></i>;<br/>


</div><b>end;</b></div><b>take </b><font color="Maroon">c<sub>7</sub></font> =  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>6</sub></font>;<br/><a NAME="E3:56_1_1_3_1_2"/><i><font color="Green">E42</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_3_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:56_1_1_3_1_2_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</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="E3:56_1_1_3_1_2_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T7">XREAL_1:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:56_1_1_3_1_2_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="seq_4.html#T10">SEQ_4:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> =  <a href="xxreal_0.html#K4">max</a> 1,<font color="Maroon">c<sub>10</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="xxreal_0.html#T16">XXREAL_0:16</a></i>;<br/>
<a NAME="E8:56_1_1_3_1_2_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E9:56_1_1_3_1_2_2"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="power.html#E6:56_1_1_3_1_2_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E10:56_1_1_3_1_2_2"/>
<font color="Maroon">c<sub>9</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="power.html#E2:56"><i><font color="Green">E38</font></i></a>, <a class="txt" href="power.html#E3:56_1_1_3_1_2_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="real_2.html#T142">REAL_2:142</a></i>;<br/>
<a NAME="E11:56_1_1_3_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E12:56_1_1_3_1_2_2"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="power.html#E9:56_1_1_3_1_2_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="real_2.html#T177">REAL_2:177</a></i>;<br/>
<a NAME="E13:56_1_1_3_1_2_2"/><i><font color="Green">E47</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E14:56_1_1_3_1_2_2"/><b>then </b><i><font color="Green">E48</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font>
 
;<br/>
<a NAME="E15:56_1_1_3_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="txt" href="power.html#E12:56_1_1_3_1_2_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="real_2.html#T178">REAL_2:178</a></i>;<br/>
<a NAME="E16:56_1_1_3_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E17:56_1_1_3_1_2_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E18:56_1_1_3_1_2_2"/><i><font color="Green">E50</font></i>: 
1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="power.html#E1:56"><i><font color="Green">E37</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E19:56_1_1_3_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="power.html#K2" target="_self">-root</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="power.html#E13:56_1_1_3_1_2_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="power.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E20:56_1_1_3_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="power.html#K1" target="_self">-root</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E21:56_1_1_3_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="power.html#K1" target="_self">-root</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</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="power.html#E17:56_1_1_3_1_2_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E22:56_1_1_3_1_2_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="power.html#K4" target="_self">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="power.html#E13:56_1_1_3_1_2_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="power.html#E18:56_1_1_3_1_2_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="power.html#T52" target="_self">Th52</a></i>;<br/>
<a NAME="E23:56_1_1_3_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="power.html#K3" target="_self">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="power.html#E2:56"><i><font color="Green">E38</font></i></a>, <a class="ref" href="real_2.html#T178">REAL_2:178</a></i>;<br/>
<a NAME="E24:56_1_1_3_1_2_2"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="power.html#K4" target="_self">to_power</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="power.html#E1:56"><i><font color="Green">E37</font></i></a>, <a class="ref" href="power.html#T37" target="_self">Th37</a></i>;<br/>
<a NAME="E25:56_1_1_3_1_2_2"/>
1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="power.html#E14:56_1_1_3_1_2_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</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="E27:56_1_1_3_1_2_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="power.html#E4:56_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="power.html#E2:56_1_1_3_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E29:56_1_1_3_1_2_2"/><i><font color="Green">E53</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="power.html#E27:56_1_1_3_1_2_2"><i><font color="Green">E52</font></i></a></i>;<br/>
<a NAME="E30:56_1_1_3_1_2_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="power.html#K4" target="_self">to_power</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="power.html#E1:56"><i><font color="Green">E37</font></i></a>, <a class="ref" href="power.html#T39" target="_self">Th39</a></i>;<br/>
<a NAME="E31:56_1_1_3_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="power.html#E29:56_1_1_3_1_2_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E32:56_1_1_3_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="power.html#E24:56_1_1_3_1_2_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E33:56_1_1_3_1_2_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="power.html#K4" target="_self">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="power.html#E1:56"><i><font color="Green">E37</font></i></a>, <a class="ref" href="power.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E34:56_1_1_3_1_2_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="power.html#E27:56_1_1_3_1_2_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E35:56_1_1_3_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="power.html#K4" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="power.html#K4" target="_self">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="power.html#E1:56"><i><font color="Green">E37</font></i></a>, <a class="txt" href="power.html#E1:56_1_1_3_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="power.html#T45" target="_self">Th45</a></i>;<br/>
<b>hence </b><a NAME="E36:56_1_1_3_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="power.html#E3:56_1_1_3_1_2_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="power.html#E33:56_1_1_3_1_2_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:56_1_1_3_1_2"/>
not <font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_3_1_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:56_1_1_3_1_2_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</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="E3:56_1_1_3_1_2_3"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</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="ref" href="xreal_1.html#T7">XREAL_1:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E5:56_1_1_3_1_2_3"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="power.html#K4" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="power.html#E1:56"><i><font color="Green">E37</font></i></a>, <a class="ref" href="power.html#T39" target="_self">Th39</a></i>;<br/>
<a NAME="E6:56_1_1_3_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="power.html#E3:56_1_1_3_1_2_3"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E7:56_1_1_3_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="power.html#E3:56_1_1_3_1_2_3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="real_2.html#T142">REAL_2:142</a></i>;<br/>
<a NAME="E8:56_1_1_3_1_2_3"/><b>then </b><i><font color="Green">E45</font></i>: 
<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>9</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  = <font color="Maroon">a<sub>1</sub></font> <a href="power.html#K1" target="_self">-root</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E10:56_1_1_3_1_2_3"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<a NAME="E11:56_1_1_3_1_2_3"/><i><font color="Green">E47</font></i>: 
1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="power.html#E1:56"><i><font color="Green">E37</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E12:56_1_1_3_1_2_3"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 holds <br/><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="power.html#K2" target="_self">-root</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="power.html#E10:56_1_1_3_1_2_3"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E13:56_1_1_3_1_2_3"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="power.html#E11:56_1_1_3_1_2_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="power.html#T24" target="_self">Th24</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E15:56_1_1_3_1_2_3"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>11</sub></font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</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>9</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="power.html#E8:56_1_1_3_1_2_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> =  <a href="xxreal_0.html#K4">max</a> 1,<font color="Maroon">c<sub>11</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="xxreal_0.html#T16">XXREAL_0:16</a></i>;<br/>
<a NAME="E17:56_1_1_3_1_2_3"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E18:56_1_1_3_1_2_3"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</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>9</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="txt" href="power.html#E15:56_1_1_3_1_2_3"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E19:56_1_1_3_1_2_3"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="power.html#K1" target="_self">-root</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</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>9</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="txt" href="power.html#E10:56_1_1_3_1_2_3"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E20:56_1_1_3_1_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="power.html#K1" target="_self">-root</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="power.html#K1" target="_self">-root</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/>
<a NAME="E21:56_1_1_3_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="power.html#K1" target="_self">-root</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</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>9</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="txt" href="power.html#E19:56_1_1_3_1_2_3"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E22:56_1_1_3_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="power.html#K1" target="_self">-root</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</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#K7">/</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E23:56_1_1_3_1_2_3"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="power.html#K4" target="_self">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="power.html#E11:56_1_1_3_1_2_3"><i><font color="Green">E47</font></i></a>, <a class="txt" href="power.html#E17:56_1_1_3_1_2_3"><i><font color="Green">E49</font></i></a>, <a class="ref" href="power.html#T52" target="_self">Th52</a></i>;<br/>
<a NAME="E24:56_1_1_3_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="power.html#K3" target="_self">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</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>9</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="power.html#E5:56_1_1_3_1_2_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E25:56_1_1_3_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="power.html#K4" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="power.html#K4" target="_self">to_power</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="power.html#K4" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="power.html#E1:56"><i><font color="Green">E37</font></i></a>, <a class="ref" href="power.html#T37" target="_self">Th37</a></i>;<br/>
<a NAME="E26:56_1_1_3_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="power.html#K4" target="_self">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="power.html#K4" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="power.html#E1:56"><i><font color="Green">E37</font></i></a>, <a class="ref" href="power.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E27:56_1_1_3_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>;<br/>
<a NAME="E28:56_1_1_3_1_2_3"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>;<br/>
<a NAME="E29:56_1_1_3_1_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="power.html#E3:56_1_1_3_1_2_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="power.html#E5:56_1_1_3_1_2_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="real_2.html#T142">REAL_2:142</a></i>;<br/>
<a NAME="E30:56_1_1_3_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="power.html#E2:56"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E31:56_1_1_3_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <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="power.html#E28:56_1_1_3_1_2_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E32:56_1_1_3_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 
;<br/>
<a NAME="E33:56_1_1_3_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="power.html#E2:56_1_1_3_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<a NAME="E34:56_1_1_3_1_2_3"/><b>then </b><i><font color="Green">E52</font></i>: 
1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i><a class="ref" href="real_2.html#T174">REAL_2:174</a></i>;<br/>
<a NAME="E35:56_1_1_3_1_2_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="power.html#E17:56_1_1_3_1_2_3"><i><font color="Green">E49</font></i></a></i>;<br/>
<b>hence </b><a NAME="E36:56_1_1_3_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="power.html#E34:56_1_1_3_1_2_3"><i><font color="Green">E52</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:56_1_1_3_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="power.html#E3:56_1_1_3_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:56_1_1"/>
ex <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="Maroon">c<sub>1</sub></font> <a href="power.html#K3" target="_self">to_power</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div>
