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

<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:26"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c2">n</font>
 ;<br/>

<a NAME="E2:26"/><i><font color="Green" title="E22">A2</font></i>: 
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1"/>
( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1"><b>suppose </b></a><a NAME="E1:26_1_1"/>
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:26_1_1"/>
(  <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font> <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="c2">n</font> &amp;  <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="idea_1.html#E2:26"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="idea_1.html#D7" target="_self" title="IDEA_1:def.7">Def7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_2"><b>suppose </b></a><a NAME="E1:26_1_2"/><i><font color="Green" title="E23">A3</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:26_1_2"/><b>then </b>
 <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font>
 <b>by </b><i><a class="ref" href="idea_1.html#D7" target="_self" title="IDEA_1:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E3:26_1_2"/>
(  <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font> <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="c2">n</font> &amp;  <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="idea_1.html#E1:26"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="idea_1.html#E1:26_1_2"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="idea_1.html#D3" target="_self" title="IDEA_1:def.3">Def3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
