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<b>let </b><font color="Maroon" title="c1">c</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:24"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c1">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 6
 ;<br/>

<a NAME="E2:24"/><i><font color="Green" title="E20">A2</font></i>: 
5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
;<br/>
<b>set </b><font color="Maroon" title="c2">i0</font> = <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c1">c</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c3">i1</font> = <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c1">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1"/>
( <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c1">c</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c1">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> or  not <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c1">c</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c1">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1"><b>suppose </b></a><a NAME="E1:24_1_1"/>
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c1">c</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c1">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>
 ;<br/><div class="add"><a NAME="E2:24_1_1"/><b>then </b>
<font color="Maroon" title="c1">c</font> is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>
 <b>by </b><i><a class="ref" href="int_1.html#T59" title="INT_1:th.59">INT_1:59</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c4">c</font> = <font color="Maroon" title="c1">c</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="asympt_1.html#E1:24"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E4:24_1_1"/><i><font color="Green" title="E21">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:24"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="asympt_1.html#E20"><i><font color="Green" title="E17">Lm13</font></i></a></i>;<br/><a NAME="E5:24_1_1"/>
<font color="Maroon" title="c4">c</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">c</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E6:24_1_1"/><b>then </b>
<font color="Maroon" title="c4">c</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 <b>by </b><i><a class="ref" href="square_1.html#T78" title="SQUARE_1:th.78">SQUARE_1:78</a></i>;<br/><b>hence </b><a NAME="E7:24_1_1"/>
<font color="Maroon" title="c1">c</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">c</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E4:24_1_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2"><b>suppose </b></a><a NAME="E1:24_1_2"/>
not <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c1">c</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c1">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>
 ;<br/><div class="add"><a NAME="E2:24_1_2"/><b>then </b><i><font color="Green" title="E21">A4</font></i>: 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c1">c</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c1">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span>
 <b>by </b><i><a class="ref" href="int_1.html#T66" title="INT_1:th.66">INT_1:66</a></i>;<br/><a NAME="E3:24_1_2"/><b>then </b><i><font color="Green" title="E22">A5</font></i>: 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c1">c</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c1">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 ;<br/><a NAME="E4:24_1_2"/><i><font color="Green" title="E23">A6</font></i>: 
<span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c1">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">c</font>
 <b>by </b><i><a class="ref" href="int_1.html#D5" title="INT_1:def.5">INT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c4">i1</font> = <span class="p1"><a href="int_1.html#K2" title="INT_1:func.2">[/</a><span class="default"><font color="Maroon" title="c1">c</font></span><a href="int_1.html#K2" title="INT_1:func.2">\]</a></span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="asympt_1.html#E1:24"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E6:24_1_2"/>
<font color="Maroon" title="c4">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 6
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:24"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="asympt_1.html#E4:24_1_2"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:24_1_2"/><b>then </b><i><font color="Green" title="E24">A7</font></i>: 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c1">c</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 5
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:24"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="asympt_1.html#E3:24_1_2"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c5">i0</font> = <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c1">c</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E9:24_1_2"/>
<font color="Maroon" title="c5">i0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 5 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="asympt_1.html#E7:24_1_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/><a NAME="E10:24_1_2"/><b>then </b><i><font color="Green" title="E25">A8</font></i>: 
<font color="Maroon" title="c4">i1</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c5">i0</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E2:24_1_2"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="asympt_1.html#E20"><i><font color="Green" title="E17">Lm13</font></i></a></i>;<br/><a NAME="E11:24_1_2"/>
<font color="Maroon" title="c4">i1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">c</font>
 <b>by </b><i><a class="ref" href="int_1.html#D5" title="INT_1:def.5">INT_1:def 5</a></i>;<br/><a NAME="E12:24_1_2"/><b>then </b>
<font color="Maroon" title="c4">i1</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">c</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 <b>by </b><i><a class="txt" href="asympt_1.html#E1:24"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></i>;<br/><a NAME="E13:24_1_2"/><b>then </b><i><font color="Green" title="E26">A9</font></i>: 
<font color="Maroon" title="c1">c</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c5">i0</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E10:24_1_2"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E14:24_1_2"/>
<font color="Maroon" title="c5">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">c</font>
 <b>by </b><i><a class="ref" href="int_1.html#D4" title="INT_1:def.4">INT_1:def 4</a></i>;<br/><a NAME="E15:24_1_2"/><b>then </b>
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c5">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">c</font>
 <b>by </b><i><a class="ref" href="pre_ff.html#T10" title="PRE_FF:th.10">PRE_FF:10</a></i>;<br/><b>hence </b><a NAME="E16:24_1_2"/>
<font color="Maroon" title="c1">c</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">c</font>
 <b>by </b><i><a class="txt" href="asympt_1.html#E13:24_1_2"><i><font color="Green" title="E26">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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