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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:33"/>
<font color="Maroon">c<sub>1</sub></font> <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:33"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="idea_1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E3:33"/><i><font color="Green">E27</font></i>: 
 <a href="idea_1.html#K7" target="_self">ChangeVal_1</a> 1,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="idea_1.html#D7" target="_self">Def7</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1"><b>suppose </b></a><a NAME="E1:33_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:33_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="idea_1.html#K7" target="_self">ChangeVal_1</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="idea_1.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E3:33_1_1"/>
2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E4:33_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="idea_1.html#K7" target="_self">ChangeVal_1</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="idea_1.html#E2:33_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1"/>
 <a href="idea_1.html#K9" target="_self">MUL_MOD</a> 1,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="idea_1.html#E3:33"><i><font color="Green">E27</font></i></a>, <a class="txt" href="idea_1.html#E1:33_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="idea_1.html#D8" target="_self">Def8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2"><b>suppose </b></a><a NAME="E1:33_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:33_1_2"/><b>then </b><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><a href="idea_1.html#K7" target="_self">ChangeVal_1</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="idea_1.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E3:33_1_2"/>
2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E4:33_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="idea_1.html#E2:33"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:33_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="idea_1.html#K7" target="_self">ChangeVal_1</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="idea_1.html#E2:33_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>;<br/><b>hence </b><a NAME="E6:33_1_2"/>
 <a href="idea_1.html#K9" target="_self">MUL_MOD</a> 1,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="idea_1.html#E3:33"><i><font color="Green">E27</font></i></a>, <a class="txt" href="idea_1.html#E2:33"><i><font color="Green">E26</font></i></a>, <a class="ref" href="idea_1.html#D8" target="_self">Def8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
