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

<a NAME="E1:32_1_1"/><i><font color="Green">E35</font></i>: 
2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n2</font> = <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">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E3:32_1_1"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">nn</font> = <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E5:32_1_1"/><i><font color="Green">E39</font></i>: 
2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i><a class="ref" href="power.html#T40">POWER:40</a></i>;<br/>
<a NAME="E6:32_1_1"/>
<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">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E7:32_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">n3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<a NAME="E9:32_1_1"/>
<font color="Maroon">n2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">nn</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n3</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<b>then </b><i><font color="Green">E40</font></i>: <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">nn</font> = 
1 <a href="nat_1.html#K6">mod</a> <font color="Maroon">nn</font>
<b>by </b><i><a class="ref" href="nat_1.html#T73">NAT_1:73</a></i>
<br/>.= 
1
<b>by </b><i>, <a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1_1_2_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">i</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="32_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:32_1_1_2_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:32_1_1_2_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>take </b><font color="Maroon">j</font> = <font color="Maroon">j</font>;<br/><i><font color="Green">E44</font></i>:  <a href="idea_1.html#K9" target="_self">MUL_MOD</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">n</font> = 
 <a href="idea_1.html#K8" target="_self">ChangeVal_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="idea_1.html#K7" target="_self">ChangeVal_1</a> 0,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">nn</font></span>)</span>,<font color="Maroon">n</font>
<b>by </b><i>, , </i>
<br/>.= 
 <a href="idea_1.html#K8" target="_self">ChangeVal_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">nn</font></span>)</span>,<font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
1
<b>by </b><i>, , </i>
;<br/><a NAME="E5:32_1_1_2_1_1"/>
<font color="Maroon">j</font> <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E6:32_1_1_2_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>(  <a href="idea_1.html#K9" target="_self">MUL_MOD</a> <font color="Maroon">i</font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:32_1_1_2_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:32_1_1_2_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:32_1_1_2_1_2"/><b>then </b>
<font color="Maroon">i</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">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:32_1_1_2_1_2"/><i><font color="Green">E46</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K6">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">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">a</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">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, , , </i>;<br/><a NAME="E6:32_1_1_2_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>;<br/><b>take </b><font color="Maroon">k</font> =  <a href="idea_1.html#K8" target="_self">ChangeVal_2</a> <font color="Maroon">a</font>,<font color="Maroon">n</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1_1_2_1_2_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:32_1_1_2_1_2_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:32_1_1_2_1_2_1_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/><a NAME="E3:32_1_1_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:32_1_1_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E1"><i><font color="Green">Lemma32</font></i></a>, <a class="txt" href="idea_1.html#E1:32"><i><font color="Green">E33</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:32_1_1_2_1_2_1_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">k</font> <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/> <a href="idea_1.html#K9" target="_self">MUL_MOD</a> <font color="Maroon">i</font>,<font color="Maroon">k</font>,<font color="Maroon">n</font> = 
 <a href="idea_1.html#K8" target="_self">ChangeVal_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="idea_1.html#K7" target="_self">ChangeVal_1</a> <font color="Maroon">k</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font>
<b>by </b><i>, </i>
<br/>.= 
 <a href="idea_1.html#K8" target="_self">ChangeVal_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font>
<b>by </b><i>, <a class="txt" href="idea_1.html#E1:32"><i><font color="Green">E33</font></i></a>, </i>
<br/>.= 
1
<b>by </b><i>, , </i>
;<br/><b>hence </b><a NAME="E7:32_1_1_2_1_2_1_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>(  <a href="idea_1.html#K9" target="_self">MUL_MOD</a> <font color="Maroon">i</font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:32_1_1_2_1_2_1_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:32_1_1_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><a NAME="E3:32_1_1_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">k</font> <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/> <a href="idea_1.html#K9" target="_self">MUL_MOD</a> <font color="Maroon">i</font>,<font color="Maroon">k</font>,<font color="Maroon">n</font> = 
 <a href="idea_1.html#K8" target="_self">ChangeVal_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="idea_1.html#K7" target="_self">ChangeVal_1</a> <font color="Maroon">k</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font>
<b>by </b><i>, </i>
<br/>.= 
 <a href="idea_1.html#K8" target="_self">ChangeVal_2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="idea_1.html#E1"><i><font color="Green">Lemma32</font></i></a>, <a class="txt" href="idea_1.html#E1:32"><i><font color="Green">E33</font></i></a>, </i>
<br/>.= 
1
<b>by </b><i>, , </i>
;<br/><b>hence </b><a NAME="E5:32_1_1_2_1_2_1_1_2"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>(  <a href="idea_1.html#K9" target="_self">MUL_MOD</a> <font color="Maroon">i</font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:32_1_1_2_1_2"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>(  <a href="idea_1.html#K9" target="_self">MUL_MOD</a> <font color="Maroon">i</font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E12:32_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>(  <a href="idea_1.html#K9" target="_self">MUL_MOD</a> <font color="Maroon">i</font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="idea_1.html#R1" target="_self">is_expressible_by</a> <font color="Maroon">n</font> )
 ;<br/>


</div>
