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<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="E1:30"/><i><font color="Green" title="E24">A1</font></i>: 
( <font color="Maroon" title="c2">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="idea_1.html#K8" title="IDEA_1:func.8">ChangeVal_2</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#K8" title="IDEA_1:func.8">ChangeVal_2</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c1">n</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1"/>
(  <a href="idea_1.html#K8" title="IDEA_1:func.8">ChangeVal_2</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="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="idea_1.html#K8" title="IDEA_1:func.8">ChangeVal_2</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>  <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="30_1_1"><b>suppose </b></a><a NAME="E1:30_1_1"/><i><font color="Green" title="E25">A2</font></i>: 
 <a href="idea_1.html#K8" title="IDEA_1:func.8">ChangeVal_2</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="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:30_1_1"/><b>then </b>
( <font color="Maroon" title="c3">a2</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 <font color="Maroon" title="c3">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="idea_1.html#E1:30"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="idea_1.html#D8" target="_self" title="IDEA_1:def.8">Def8</a></i>;<br/><b>hence </b><a NAME="E3:30_1_1"/>
<font color="Maroon" title="c3">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a1</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E1:30"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="idea_1.html#E1:30_1_1"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="idea_1.html#D8" target="_self" title="IDEA_1:def.8">Def8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2"><b>suppose </b></a><a NAME="E1:30_1_2"/><i><font color="Green" title="E25">A3</font></i>: 
 <a href="idea_1.html#K8" title="IDEA_1:func.8">ChangeVal_2</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:30_1_2"/><b>then </b><i><font color="Green" title="E26">A4</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#D8" target="_self" title="IDEA_1:def.8">Def8</a></i>;<br/><a NAME="E3:30_1_2"/>
<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#E1:30"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="idea_1.html#E1:30_1_2"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="idea_1.html#D8" target="_self" title="IDEA_1:def.8">Def8</a></i>;<br/><b>hence </b><font color="Maroon" title="c3">a2</font> = 
 <a href="idea_1.html#K8" title="IDEA_1:func.8">ChangeVal_2</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#E1:30"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="idea_1.html#D8" target="_self" title="IDEA_1:def.8">Def8</a></i>
<br/>.= 
<font color="Maroon" title="c2">a1</font>
<b>by </b><i><a class="txt" href="idea_1.html#E2:30_1_2"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="idea_1.html#D8" target="_self" title="IDEA_1:def.8">Def8</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

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