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

<b>let </b><font color="Maroon" title="c1">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>let </b><font color="Maroon" title="c2">a1</font>, <font color="Maroon" title="c3">a2</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="E1:27"/><i><font color="Green" title="E22">A1</font></i>: 
( <font color="Maroon" title="c2">a1</font> <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c1">n</font> )
 ;<br/>

<a NAME="E2:27"/><b>then </b><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon" title="c2">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="idea_1.html#D3" target="_self" title="IDEA_1:def.3">Def3</a></i>;<br/>
<a NAME="E3:27"/><i><font color="Green" title="E24">A3</font></i>: 
<font color="Maroon" title="c3">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="idea_1.html#E1:27"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="idea_1.html#D3" target="_self" title="IDEA_1:def.3">Def3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1"/>
(  <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font> or  <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1"><b>suppose </b></a><a NAME="E1:27_1_1"/><i><font color="Green" title="E25">A4</font></i>: 
 <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c3">a2</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="idea_1.html#E1:27"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="idea_1.html#E3:27"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="idea_1.html#D7" target="_self" title="IDEA_1:def.7">Def7</a></i>
<br/>.= 
<font color="Maroon" title="c2">a1</font>
<b>by </b><i><a class="txt" href="idea_1.html#E2:27"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="idea_1.html#E1:27_1_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="idea_1.html#D7" target="_self" title="IDEA_1:def.7">Def7</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_2"><b>suppose </b></a><a NAME="E1:27_1_2"/><i><font color="Green" title="E25">A5</font></i>: 
 <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c3">a2</font> = 
 <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="idea_1.html#E1:27"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="idea_1.html#D7" target="_self" title="IDEA_1:def.7">Def7</a></i>
<br/>.= 
<font color="Maroon" title="c2">a1</font>
<b>by </b><i><a class="txt" href="idea_1.html#E1:27_1_2"><i><font color="Green" title="E25">A5</font></i></a>, <a class="ref" href="idea_1.html#D7" target="_self" title="IDEA_1:def.7">Def7</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
