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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">m</font>, <font color="Olive">k</font>, <font color="Olive">f</font>, <font color="Olive">ie</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">ie</font> <a href="radix_1.html#R1">is_represented_by</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Olive">k</font> &amp; <font color="Olive">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/>for <font color="Olive">e</font> being  <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">a<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a>   st <font color="Olive">e</font> <a href="hidden.html#R1">=</a>  <a href="radix_2.html#K5" target="_self">DecSD2</a> <font color="Olive">ie</font>,<font color="Maroon">a<sub>1</sub></font>,<font color="Olive">k</font> holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Olive">m</font>,<font color="Olive">e</font>,<font color="Olive">f</font>,<font color="Olive">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">ie</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Olive">f</font>;<br/>
<a NAME="E1:26"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">f</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">ie</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:26_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">ie</font> <a href="radix_1.html#R1">is_represented_by</a> 1,<font color="Maroon">k</font> &amp; <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<b>let </b><font color="Maroon">e</font> be   <a href="binarith.html#NM1">Tuple</a> of 1,<a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E2:26_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="radix_2.html#K5" target="_self">DecSD2</a> <font color="Maroon">ie</font>,1,<font color="Maroon">k</font>
 ;<br/>

<a NAME="E3:26_1"/>
<font color="Maroon">ie</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> 1
 
<b>by </b><i>, <a class="ref" href="radix_1.html#D12">RADIX_1:def 12</a></i>;<br/>
<a NAME="E4:26_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">ie</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>;<br/>
<a NAME="E5:26_1"/><i><font color="Green">E36</font></i>: 
1 <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> 0
 
;<br/>
<a NAME="E6:26_1"/><i><font color="Green">E47</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:26_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="radix_2.html#K5" target="_self">DecSD2</a> <font color="Maroon">ie</font>,1,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E8:26_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="radix_2.html#K5" target="_self">DecSD2</a> <font color="Maroon">ie</font>,1,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="radix_2.html#T2" target="_self">Th2</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then </b><font color="Maroon">e</font> <a href="finseq_4.html#K4">/.</a> 1 = 
<span class="p1">(<span class="default"><a href="radix_2.html#K5" target="_self">DecSD2</a> <font color="Maroon">ie</font>,1,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> 1
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="radix_2.html#K4" target="_self">DigitDC2</a> <font color="Maroon">ie</font>,1,<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="radix_2.html#T2" target="_self">Th2</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">ie</font> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> 0</span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">ie</font> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> 1
<b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>
<br/>.= 
<font color="Maroon">ie</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="nat_2.html#T6">NAT_2:6</a></i>
<br/>.= 
<font color="Maroon">ie</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>
<br/>.= 
<font color="Maroon">ie</font>
<b>by </b><i>, <a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>
;<br/>
<a NAME="E10:26_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">ie</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="radix_2.html#K8" target="_self">Table2</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>,1
 
;<br/>
<b>hence </b><a NAME="E11:26_1"/>
<span class="p1">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">ie</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:26"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:26_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>] )
 ;<br/>

<a NAME="E2:26_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">f</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">ie</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E3:26_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">ie</font> <a href="radix_1.html#R1">is_represented_by</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1,<font color="Maroon">k</font> &amp; <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<b>let </b><font color="Maroon">e</font> be   <a href="binarith.html#NM1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E4:26_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="radix_2.html#K5" target="_self">DecSD2</a> <font color="Maroon">ie</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  = <font color="Maroon">e</font> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<b>consider </b><font color="Maroon">en</font> being    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:26_2"/><i><font color="Green">E53</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">en</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E7:26_2"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">en</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">en</font> = <font color="Maroon">en</font> as   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">n</font>,<a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E9:26_2"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E10:26_2"/><b>then </b><i><font color="Green">E55</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T9">FINSEQ_2:9</a></i>;<br/>
<a NAME="E11:26_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E12:26_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E13:26_2"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">en</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> implies <span class="p1">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">en</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E1:26_2_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E2:26_2_1"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:26_2_1_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> implies <span class="p1">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">en</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> )
 ;<br/>

<b>assume </b><a NAME="E2:26_2_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E3:26_2_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:26_2_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <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/>
<a NAME="E5:26_2_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
;<br/>
<a NAME="E6:26_2_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="nat_1.html#K1">+</a> 0 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E7:26_2_1_1"/><i><font color="Green">E74</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">en</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:26_2_1_1"/><i><font color="Green">E75</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">e</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:26_2_1_1"/><b>then </b><i><font color="Green">E76</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_2_1_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:26_2_1_1_1_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">en</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="radix_2.html#K8" target="_self">Table2</a> <font color="Maroon">m</font>,<font color="Maroon">en</font>,<font color="Maroon">f</font>,<font color="Maroon">n</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">en</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="radix_2.html#T4" target="_self">Th4</a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="wsierp_1.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font>
<b>by </b><i>, <a class="ref" href="radix_2.html#T4" target="_self">Th4</a></i>
<br/>.= 
 <a href="radix_2.html#K8" target="_self">Table2</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> 1
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E3:26_2_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">en</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="radix_2.html#T5" target="_self">Th5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:26_2_1_1_1_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">nn</font> = <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E3:26_2_1_1_1_1_2"/><i><font color="Green">E79</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">nn</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, <a class="ref" href="radix_2.html#T5" target="_self">Th5</a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E4:26_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E80</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 )
 ;<br/><a NAME="E5:26_2_1_1_1_1_2"/><i><font color="Green">E81</font></i>: 
 ex <font color="Olive">I1</font>, <font color="Olive">I2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">I1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">en</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">I2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">en</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Olive">I2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">I1</font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K8" target="_self">Table2</a> <font color="Maroon">m</font>,<font color="Maroon">en</font>,<font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font> )
 <b>by </b><i>, , <a class="ref" href="radix_2.html#T5" target="_self">Th5</a>, </i>;<br/><a NAME="E6:26_2_1_1_1_1_2"/><i><font color="Green">E82</font></i>: 
 ex <font color="Olive">I1</font>, <font color="Olive">I2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">I1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">I2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Olive">I2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">I1</font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K8" target="_self">Table2</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font> )
 <b>by </b><i>, <a class="ref" href="radix_2.html#T7" target="_self">Th7</a>, </i>;<br/><a NAME="E7:26_2_1_1_1_1_2"/>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E8:26_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E83</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="radix_2.html#T6" target="_self">Th6</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E9:26_2_1_1_1_1_2"/>
<font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="binarith.html#T52">BINARITH:52</a></i>;<br/><a NAME="E10:26_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="radix_2.html#T9" target="_self">Th9</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E11:26_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">en</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E12:26_2_1_1_1_1_2"/><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T52">BINARITH:52</a></i>;<br/><a NAME="E13:26_2_1_1_1_1_2"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E14:26_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <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/><a NAME="E15:26_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E87</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">nn</font>
 <b>by </b><i><a class="ref" href="jordan3.html#T4">JORDAN3:4</a></i>;<br/><a NAME="E16:26_2_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">nn</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">nn</font>
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E17:26_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E88</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="radix_2.html#E1:26_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E18:26_2_1_1_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="radix_2.html#E2:26_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E19:26_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="radix_2.html#E2:26"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E20:26_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E89</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/><span class="p1">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">en</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><font color="Maroon">en</font> <a href="wsierp_1.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font>
<b>by </b><i>, <a class="ref" href="radix_2.html#T6" target="_self">Th6</a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><font color="Maroon">e</font> <a href="wsierp_1.html#K3">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><font color="Maroon">e</font> <a href="wsierp_1.html#K3">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="radix_2.html#T6" target="_self">Th6</a>, <a class="ref" href="binarith.html#T56">BINARITH:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K8" target="_self">Table2</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font>
<b>by </b><i><a class="txt" href="radix_2.html#E3:26_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E22:26_2_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">en</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="radix_2.html#T8" target="_self">Th8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:26_2_1_1"/>
<span class="p1">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">en</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:26_2_1"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E4:26_2_1"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">en</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E14:26_2"/>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E15:26_2"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<b>set </b><font color="Maroon">ien</font> =  <a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font>;<br/>
<a NAME="E16:26_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">ie</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">r2</font> =  <a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  = <span class="p1">(<span class="default"><a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">rD</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:26_2_2"/><i><font color="Green">E92</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">rD</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E4:26_2_2"/><i><font color="Green">E93</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">rD</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">i</font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">rD</font> = <font color="Maroon">rD</font> as   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">n</font>,<a href="numbers.html#K1">REAL</a>  <b>by </b><i>, <a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<i><font color="Green">E94</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
;<br/>
<a NAME="E7:26_2_2"/><i><font color="Green">E95</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">rD</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:26_2_2"/><i><font color="Green">E131</font></i>: 
1 <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> 0
 
;<br/>
<a NAME="E9:26_2_2"/><i><font color="Green">E132</font></i>: 
 <a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">e</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="radix_2.html#K1" target="_self">SubDigit2</a> <font color="Maroon">e</font>,1,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="euler_2.html#K1">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:26_2_2_2"/><i><font color="Green">E133</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">e</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<i><font color="Green">E134</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="euler_2.html#K1">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">rD</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>
<br/>.= 
<font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
;<br/>
<a NAME="E3:26_2_2_2"/><i><font color="Green">E135</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="radix_2.html#K1" target="_self">SubDigit2</a> <font color="Maroon">e</font>,1,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<i><font color="Green">E136</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" target="_self">SubDigit2</a> <font color="Maroon">e</font>,1,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="euler_2.html#K1">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" target="_self">SubDigit2</a> <font color="Maroon">e</font>,1,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="euler_2.html#K1">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
;<br/>
<a NAME="E5:26_2_2_2"/><b>then </b><i><font color="Green">E137</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">e</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" target="_self">SubDigit2</a> <font color="Maroon">e</font>,1,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="euler_2.html#K1">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E6:26_2_2_2"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">e</font>,<font color="Maroon">k</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">e</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" target="_self">SubDigit2</a> <font color="Maroon">e</font>,1,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="euler_2.html#K1">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_2_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:26_2_2_2_3"/><i><font color="Green">E138</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">e</font>,<font color="Maroon">k</font></span>)</span> )
 ;<br/>

<a NAME="E2:26_2_2_2_3"/><b>then </b><i><font color="Green">E139</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E3:26_2_2_2_3"/><b>then </b><i><font color="Green">E140</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:26_2_2_2_3"/><b>then </b><i><font color="Green">E141</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">e</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_2_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_2_2_2_3_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_2_2_3_1_1"><b>suppose </b></a><a NAME="E1:26_2_2_2_3_1_1"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" target="_self">SubDigit2</a> <font color="Maroon">e</font>,1,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="euler_2.html#K1">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> = 
 <a href="radix_2.html#K1" target="_self">SubDigit2</a> <font color="Maroon">e</font>,1,<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">e</font>,<font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="ref" href="radix_2.html#T3" target="_self">Th3</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">e</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> 1
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E3:26_2_2_2_3_1_1"/>
<span class="p1">(<span class="default"><a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">e</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" target="_self">SubDigit2</a> <font color="Maroon">e</font>,1,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="euler_2.html#K1">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_2_2_3_1_2"><b>suppose </b></a><a NAME="E1:26_2_2_2_3_1_2"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:26_2_2_2_3_1_2"/>
1 <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <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">i1</font> = <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M2">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="E4:26_2_2_2_3_1_2"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:26_2_2_2_3_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E6:26_2_2_2_3_1_2"/><b>then </b><i><font color="Green">E144</font></i>: 
<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 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <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/><a NAME="E7:26_2_2_2_3_1_2"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <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="radix_2.html#T8" target="_self">Th8</a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E8:26_2_2_2_3_1_2"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="radix_2.html#E2:26"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E9:26_2_2_2_3_1_2"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">rD</font>
 <b>by </b><i>, <a class="txt" href="radix_2.html#E2:26"><i><font color="Green">E48</font></i></a>, <a class="txt" href="radix_2.html#E1:26_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>then reconsider </b><font color="Maroon">rs</font> = <font color="Maroon">rD</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/><b>reconsider </b><font color="Maroon">rs2</font> = <span class="p1">(<span class="default"><a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, , <a class="txt" href="radix_2.html#E3:26_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/><a NAME="E12:26_2_2_2_3_1_2"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" target="_self">SubDigit2</a> <font color="Maroon">e</font>,1,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="euler_2.html#K1">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="euler_2.html#K1">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="radix_2.html#T5" target="_self">Th5</a>, <a class="ref" href="radix_2.html#T6" target="_self">Th6</a>, <a class="ref" href="radix_2.html#T8" target="_self">Th8</a>, <a class="ref" href="finseq_1.html#T37">FINSEQ_1:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">rD</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i>, , , <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>
<br/>.= 
<font color="Maroon">r2</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">rs</font>
<b>by </b><i><a class="txt" href="radix_2.html#E2:26_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="rvsum_1.html#T67">RVSUM_1:67</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">rs2</font>
<b>by </b><i>, , , <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span>
<b>by </b><i>, , <a class="txt" href="radix_2.html#E3:26_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="radix_2.html#K1" target="_self">SubDigit2</a> <font color="Maroon">en</font>,<font color="Maroon">i1</font>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_2.html#E2:26_2"><i><font color="Green">E50</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">en</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i1</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">en</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i1</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">i1</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">en</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i1</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_2.html#E2:26"><i><font color="Green">E48</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">i1</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="wsierp_1.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="radix_2.html#E2:26_2"><i><font color="Green">E50</font></i></a></i>
<br/>.= 
 <a href="radix_2.html#K1" target="_self">SubDigit2</a> <font color="Maroon">e</font>,<font color="Maroon">i</font>,<font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">e</font>,<font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="radix_2.html#T9" target="_self">Th9</a>, </i>
;<br/><b>hence </b><a NAME="E14:26_2_2_2_3_1_2"/>
<span class="p1">(<span class="default"><a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">e</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" target="_self">SubDigit2</a> <font color="Maroon">e</font>,1,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="euler_2.html#K1">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E7:26_2_2_2"/>
 <a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">e</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="radix_2.html#K1" target="_self">SubDigit2</a> <font color="Maroon">e</font>,1,<font color="Maroon">k</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="wsierp_1.html#K4">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="euler_2.html#K1">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="radix_2.html#T7" target="_self">Th7</a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:26_2_2"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">e</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E11:26_2_2"/><b>then </b><i><font color="Green">E150</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">e</font>
 
<b>by </b><i><a class="ref" href="radix_2.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<font color="Maroon">ie</font> = 
 <a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">e</font>,<font color="Maroon">k</font>
<b>by </b><i>, , <a class="ref" href="radix_2.html#T2" target="_self">Th2</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K1" target="_self">SubDigit2</a> <font color="Maroon">e</font>,1,<font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="euler_2.html#K1">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" target="_self">DigitSD2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T106">RVSUM_1:106</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K1" target="_self">SubDigit2</a> <font color="Maroon">e</font>,1,<font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">rD</font></span>)</span></span>)</span>
<b>by </b><i>, , , <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K1" target="_self">SubDigit2</a> <font color="Maroon">e</font>,1,<font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">rD</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T117">RVSUM_1:117</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K1" target="_self">SubDigit2</a> <font color="Maroon">e</font>,1,<font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, , , <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> 0</span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>
;<br/>
<b>hence </b><a NAME="E13:26_2_2"/>
<font color="Maroon">ie</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:26_2"/><b>then </b><i><font color="Green">E151</font></i>: 
<font color="Maroon">ie</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="int_1.html#T19">INT_1:19</a></i>;<br/>
<a NAME="E18:26_2"/><i><font color="Green">E152</font></i>: 
 <a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E19:26_2"/>
<font color="Maroon">ie</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="radix_1.html#D12">RADIX_1:def 12</a></i>;<br/>
<a NAME="E20:26_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E21:26_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/>
<a NAME="E22:26_2"/><b>then </b>
 <a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E23:26_2"/><b>then </b><i><font color="Green">E153</font></i>: 
 <a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font> <a href="radix_1.html#R1">is_represented_by</a> <font color="Maroon">n</font>,<font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="radix_1.html#D12">RADIX_1:def 12</a></i>;<br/>
<a NAME="E24:26_2"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">en</font> <a href="hidden.html#R1">=</a>  <a href="radix_2.html#K5" target="_self">DecSD2</a> <span class="p1">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:26_2_3"/><i><font color="Green">E155</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">en</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="radix_2.html#K5" target="_self">DecSD2</a> <span class="p2">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E2:26_2_3"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">en</font> holds <br/><font color="Maroon">en</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K5" target="_self">DecSD2</a> <span class="p2">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:26_2_3_1"/><i><font color="Green">E156</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">en</font> )
 ;<br/>

<a NAME="E2:26_2_3_1"/><b>then </b><i><font color="Green">E157</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:26_2_3_1"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <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="E4:26_2_3_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E5:26_2_3_1"/><b>then </b><i><font color="Green">E159</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<font color="Maroon">e</font> <a href="wsierp_1.html#K3">.</a> 1 = 
 <a href="radix_2.html#K4" target="_self">DigitDC2</a> <font color="Maroon">ie</font>,1,<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="radix_2.html#T2" target="_self">Th2</a>, <a class="ref" href="radix_2.html#T3" target="_self">Th3</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">ie</font> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">ie</font> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> 1
<b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>
<br/>.= 
<font color="Maroon">ie</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="nat_2.html#T6">NAT_2:6</a></i>
<br/>.= 
<font color="Maroon">ie</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>
;<br/>
<a NAME="E7:26_2_3_1"/><b>then </b><i><font color="Green">E161</font></i>: 
<font color="Maroon">e</font> <a href="wsierp_1.html#K3">.</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/>
<a NAME="E8:26_2_3_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">e</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:26_2_3_1"/><b>then </b><i><font color="Green">E162</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E10:26_2_3_1"/><i><font color="Green">E163</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font> )
 
<b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:26_2_3_1"/><i><font color="Green">E164</font></i>: 
(  <a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">e</font> <a href="wsierp_1.html#K3">.</a> 1 as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon">b</font> =  <a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<i><font color="Green">E170</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> = 
<font color="Maroon">G</font> <a href="int_1.html#K5">div</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">e</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="ref" href="int_1.html#D7">INT_1:def 7</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">e</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T114">XCMPLX_1:114</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="int_1.html#T51">INT_1:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="int_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="int_1.html#D7">INT_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>
<br/>.= 
0 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="radix_2.html#T6" target="_self">Th6</a>, <a class="ref" href="nat_1.html#T80">NAT_1:80</a></i>
;<br/>
<font color="Maroon">en</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">e</font> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="radix_2.html#K4" target="_self">DigitDC2</a> <font color="Maroon">ie</font>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="radix_2.html#T2" target="_self">Th2</a>, <a class="ref" href="radix_2.html#T5" target="_self">Th5</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="radix_2.html#T3" target="_self">Th3</a>, <a class="ref" href="radix_2.html#T7" target="_self">Th7</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="radix_2.html#T8" target="_self">Th8</a>, <a class="ref" href="pepin.html#T27">PEPIN:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">e</font> <a href="wsierp_1.html#K3">.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="nat_2.html#T29">NAT_2:29</a></i>
<br/>.= 
 <a href="radix_2.html#K4" target="_self">DigitDC2</a> <span class="p1">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font>,<font color="Maroon">k</font>
<b>by </b><i>, , <a class="ref" href="radix_2.html#T9" target="_self">Th9</a>, </i>
;<br/>
<b>hence </b><a NAME="E17:26_2_3_1"/>
<font color="Maroon">en</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K5" target="_self">DecSD2</a> <span class="p2">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:26_2_3"/>
<font color="Maroon">en</font> <a href="hidden.html#R1">=</a>  <a href="radix_2.html#K5" target="_self">DecSD2</a> <span class="p1">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">n</font>,<font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E25:26_2"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
;<br/>
<a NAME="E26:26_2"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E27:26_2"/><b>then </b>
 ex <font color="Olive">I1</font>, <font color="Olive">I2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">I1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">n</font> &amp; <font color="Olive">I2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Olive">I2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">I1</font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K8" target="_self">Table2</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">en</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K8" target="_self">Table2</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font>
<b>by </b><i><a class="txt" href="radix_2.html#E1:26"><i><font color="Green">E29</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K8" target="_self">Table2</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K8" target="_self">Table2</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="pepin.html#T12">PEPIN:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K8" target="_self">Table2</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="euler_2.html#T10">EULER_2:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K8" target="_self">Table2</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="newton.html#T14">NEWTON:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><font color="Maroon">e</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="euler_2.html#T9">EULER_2:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_2.html#K3" target="_self">SDDec2</a> <font color="Maroon">en</font>,<font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>
;<br/>
<b>hence </b><a NAME="E29:26_2"/>
<span class="p1">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">ie</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:26"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S9">NAT_1:sch 9</a>(, );<br/></i>
<b>hence </b><a NAME="E4:26"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 holds <br/>for <font color="Olive">m</font>, <font color="Olive">k</font>, <font color="Olive">f</font>, <font color="Olive">ie</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">ie</font> <a href="radix_1.html#R1">is_represented_by</a> <font color="Olive">n</font>,<font color="Olive">k</font> &amp; <font color="Olive">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/>for <font color="Olive">e</font> being  <a href="binarith.html#NM1">Tuple</a> of <font color="Olive">n</font>,<a href="numbers.html#K5">NAT</a>   st <font color="Olive">e</font> <a href="hidden.html#R1">=</a>  <a href="radix_2.html#K5" target="_self">DecSD2</a> <font color="Olive">ie</font>,<font color="Olive">n</font>,<font color="Olive">k</font> holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K9" target="_self">Pow_mod</a> <font color="Olive">m</font>,<font color="Olive">e</font>,<font color="Olive">f</font>,<font color="Olive">k</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">ie</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Olive">f</font>
 ;<br/>


</div>
