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<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font>, <font color="Maroon" title="c3">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/>





<a NAME="E1:35"/><i><font color="Green" title="E28">A1</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></i>;<br/>
<a NAME="E2:35"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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="c3">n</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> <font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_d.html#T1" title="NAT_D:th.1">NAT_D:1</a></i>;<br/>
<a NAME="E3:35"/><b>then </b><i><font color="Green" title="E29">A2</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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="c3">n</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> <font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="c3">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">CV</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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="c3">n</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> <font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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="c3">n</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> <font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="c3">n</font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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="c3">n</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> <font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font> )
 </span><b>by </b><i><a class="txt" href="idea_1.html#E3:35"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1"><b>suppose </b></a><a NAME="E1:35_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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="c3">n</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> <font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="c3">n</font>
 ;<br/><div class="add"><a NAME="E2:35_1_1_1"/><b>then </b>
 <a href="idea_1.html#K8" title="IDEA_1:func.8">ChangeVal_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="c3">n</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> <font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c3">n</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="ref" href="idea_1.html#D8" target="_self" title="IDEA_1:def.8">Def8</a></i>;<br/><b>hence </b><a NAME="E3:35_1_1_1"/>
 <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c3">n</font> <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E1:35"><i><font color="Green" title="E28">A1</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2"><b>suppose </b></a><a NAME="E1:35_1_1_2"/><i><font color="Green" title="E30">A3</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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="c3">n</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> <font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font>
 ;<br/><div class="add"><a NAME="E2:35_1_1_2"/><b>then </b>
 <a href="idea_1.html#K8" title="IDEA_1:func.8">ChangeVal_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="c3">n</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> <font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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="c3">n</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> <font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="idea_1.html#D8" target="_self" title="IDEA_1:def.8">Def8</a></i>;<br/><b>hence </b><a NAME="E3:35_1_1_2"/>
 <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c3">n</font> <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E1:35_1_1_2"><i><font color="Green" title="E30">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><b>end;</b></div>
<b>hence </b><a NAME="E5:35"/>
 <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c3">n</font> <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c3">n</font>
 ;<br/>


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