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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">k</font>, <font color="Olive" title="b3">f</font>, <font color="Olive" title="b4">ie</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b4">ie</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> $1,<font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b3">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 holds <br/>for <font color="Olive" title="b5">e</font> being  <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of $1,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <font color="Olive" title="b4">ie</font>,$1,<font color="Olive" title="b2">k</font> holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Olive" title="b1">m</font>,<font color="Olive" title="b5">e</font>,<font color="Olive" title="b3">f</font>,<font color="Olive" title="b2">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Olive" title="b4">ie</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Olive" title="b3">f</font>;<br/>
<a NAME="E1:26"/><i><font color="Green" title="E18">A1</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" title="c1">m</font>, <font color="Maroon" title="c2">k</font>, <font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">ie</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:26_1"/><i><font color="Green" title="E19">A2</font></i>: 
( <font color="Maroon" title="c4">ie</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> 1,<font color="Maroon" title="c2">k</font> &amp; <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 ;<br/>

<b>let </b><font color="Maroon" title="c5">e</font> be   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of 1,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E2:26_1"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <font color="Maroon" title="c4">ie</font>,1,<font color="Maroon" title="c2">k</font>
 ;<br/>

<a NAME="E3:26_1"/>
<font color="Maroon" title="c4">ie</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 1
 
<b>by </b><i><a class="txt" href="radix_2.html#E1:26_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="radix_1.html#D12" title="RADIX_1:def.12">RADIX_1:def 12</a></i>;<br/>
<a NAME="E4:26_1"/><b>then </b><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon" title="c4">ie</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font>
 
<b>by </b><i><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></i>;<br/>
<a NAME="E5:26_1"/><i><font color="Green" title="E22">A5</font></i>: 
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
;<br/>
<a NAME="E6:26_1"/><i><font color="Green" title="E23">A6</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:26_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <font color="Maroon" title="c4">ie</font>,1,<font color="Maroon" title="c2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E8:26_1"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <font color="Maroon" title="c4">ie</font>,1,<font color="Maroon" title="c2">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="radix_2.html#E6:26_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then </b><font color="Maroon" title="c5">e</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
<span class="p1">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <font color="Maroon" title="c4">ie</font>,1,<font color="Maroon" title="c2">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> 1
<b>by </b><i><a class="txt" href="radix_2.html#E2:26_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="radix_2.html#K4" title="RADIX_2:func.4">DigitDC2</a> <font color="Maroon" title="c4">ie</font>,1,<font color="Maroon" title="c2">k</font>
<b>by </b><i><a class="txt" href="radix_2.html#E6:26_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="radix_2.html#D5" target="_self" title="RADIX_2:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">ie</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="radix_2.html#E5:26_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">ie</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 1
<b>by </b><i><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>
<br/>.= 
<font color="Maroon" title="c4">ie</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="nat_2.html#T6" title="NAT_2:th.6">NAT_2:6</a></i>
<br/>.= 
<font color="Maroon" title="c4">ie</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></i>
<br/>.= 
<font color="Maroon" title="c4">ie</font>
<b>by </b><i><a class="txt" href="radix_2.html#E4:26_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="nat_d.html#T24" title="NAT_D:th.24">NAT_D:24</a></i>
;<br/>
<a NAME="E10:26_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">ie</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_2.html#K8" title="RADIX_2:func.8">Table2</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c3">f</font>,1
 
;<br/>
<b>hence </b><a NAME="E11:26_1"/>
<span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">ie</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="ref" href="radix_2.html#D9" target="_self" title="RADIX_2:def.9">Def9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:26"/><i><font color="Green" title="E19">A7</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:26_2"/><i><font color="Green" title="E20">A8</font></i>: 
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font>] )
 ;<br/>

<a NAME="E2:26_2"/><i><font color="Green" title="E21">A9</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<b>let </b><font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">k</font>, <font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">ie</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E3:26_2"/><i><font color="Green" title="E22">A10</font></i>: 
( <font color="Maroon" title="c5">ie</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 ;<br/>

<b>let </b><font color="Maroon" title="c6">e</font> be   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E4:26_2"/><i><font color="Green" title="E23">A11</font></i>: 
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <font color="Maroon" title="c5">ie</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  = <font color="Maroon" title="c6">e</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c7">en</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:26_2"/><i><font color="Green" title="E24">A12</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">en</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>and </b><br/><a NAME="E7:26_2"/><i><font color="Green" title="E25">A13</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c7">en</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c8">en</font> = <font color="Maroon" title="c7">en</font> as   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c1">n</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="radix_2.html#E6:26_2"><i><font color="Green" title="E24">A12</font></i></a>, <a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E9:26_2"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E10:26_2"/><b>then </b><i><font color="Green" title="E26">A14</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T9" title="FINSEQ_2:th.9">FINSEQ_2:9</a></i>;<br/>
<a NAME="E11:26_2"/><i><font color="Green" title="E27">A15</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="radix_2.html#E1:26_2"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E12:26_2"/><i><font color="Green" title="E28">A16</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="radix_2.html#E2:26_2"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E13:26_2"/><i><font color="Green" title="E29">A17</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Olive" title="b1">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" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> implies <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> $1 );<br/>
<a NAME="E1:26_2_1"/><i><font color="Green" title="E30">A18</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E2:26_2_1"/><i><font color="Green" title="E31">A19</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="c9">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:26_2_1_1"/><i><font color="Green" title="E32">A20</font></i>: 
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> implies <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c9">i</font> )
 ;<br/>

<b>assume </b><a NAME="E2:26_2_1_1"/><i><font color="Green" title="E33">A21</font></i>: 
<font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<a NAME="E3:26_2_1_1"/><b>then </b><i><font color="Green" title="E34">A22</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:26_2_1_1"/><b>then </b><i><font color="Green" title="E35">A23</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E5:26_2_1_1"/>
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 
;<br/>
<a NAME="E6:26_2_1_1"/><b>then </b><i><font color="Green" title="E36">A24</font></i>: 
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E7:26_2_1_1"/><i><font color="Green" title="E37">A25</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">en</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="radix_2.html#E6:26_2"><i><font color="Green" title="E24">A12</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:26_2_1_1"/><i><font color="Green" title="E38">A26</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">e</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:26_2_1_1"/><b>then </b><i><font color="Green" title="E39">A27</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">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" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 </span><b>by </b><i><a class="txt" href="radix_2.html#E6:26_2_1_1"><i><font color="Green" title="E36">A24</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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" title="E40">A28</font></i>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
 <a href="radix_2.html#K8" title="RADIX_2:func.8">Table2</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="radix_2.html#E1:26_2"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="radix_2.html#D9" target="_self" title="RADIX_2:def.9">Def9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">en</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
<b>by </b><i><a class="txt" href="radix_2.html#E11:26_2"><i><font color="Green" title="E27">A15</font></i></a>, <a class="txt" href="radix_2.html#E7:26_2_1_1"><i><font color="Green" title="E37">A25</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
<b>by </b><i><a class="txt" href="radix_2.html#E7:26_2"><i><font color="Green" title="E25">A13</font></i></a>, <a class="txt" href="radix_2.html#E11:26_2"><i><font color="Green" title="E27">A15</font></i></a></i>
<br/>.= 
 <a href="radix_2.html#K8" title="RADIX_2:func.8">Table2</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="radix_2.html#E12:26_2"><i><font color="Green" title="E28">A16</font></i></a>, <a class="txt" href="radix_2.html#E9:26_2_1_1"><i><font color="Green" title="E39">A27</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> 1
<b>by </b><i><a class="txt" href="radix_2.html#E2:26_2"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="radix_2.html#D9" target="_self" title="RADIX_2:def.9">Def9</a></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" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="radix_2.html#E1:26_2_1_1_1_1_1"><i><font color="Green" title="E40">A28</font></i></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" title="E40">A29</font></i>: 
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c10">nn</font> = <font color="Maroon" title="c1">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="radix_2.html#E1:26_2"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E3:26_2_1_1_1_1_2"/><i><font color="Green" title="E41">A30</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">nn</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="radix_2.html#E4:26_2_1_1"><i><font color="Green" title="E35">A23</font></i></a>, <a class="txt" href="radix_2.html#E1:26_2_1_1_1_1_2"><i><font color="Green" title="E40">A29</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E4:26_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E42">A31</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 )
 ;<br/><a NAME="E5:26_2_1_1_1_1_2"/><i><font color="Green" title="E43">A32</font></i>: 
 ex <font color="Olive" title="b1">I1</font>, <font color="Olive" title="b2">I2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">I1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Olive" title="b2">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">I1</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K8" title="RADIX_2:func.8">Table2</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c4">f</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">i</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font> )
 <b>by </b><i><a class="txt" href="radix_2.html#E1:26_2"><i><font color="Green" title="E20">A8</font></i></a>, <a class="txt" href="radix_2.html#E4:26_2_1_1"><i><font color="Green" title="E35">A23</font></i></a>, <a class="txt" href="radix_2.html#E1:26_2_1_1_1_1_2"><i><font color="Green" title="E40">A29</font></i></a>, <a class="ref" href="radix_2.html#D9" target="_self" title="RADIX_2:def.9">Def9</a></i>;<br/><a NAME="E6:26_2_1_1_1_1_2"/><i><font color="Green" title="E44">A33</font></i>: 
 ex <font color="Olive" title="b1">I1</font>, <font color="Olive" title="b2">I2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">I1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Olive" title="b2">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">I1</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K8" title="RADIX_2:func.8">Table2</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">i</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font> )
 <b>by </b><i><a class="txt" href="radix_2.html#E2:26_2"><i><font color="Green" title="E21">A9</font></i></a>, <a class="txt" href="radix_2.html#E4:26_2_1_1_1_1_2"><i><font color="Green" title="E42">A31</font></i></a>, <a class="ref" href="radix_2.html#D9" target="_self" title="RADIX_2:def.9">Def9</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" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c9">i</font>
 <b>by </b><i><a class="txt" href="radix_2.html#E3:26_2_1_1"><i><font color="Green" title="E34">A22</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E8:26_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E45">A34</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">i</font>
 <b>by </b><i><a class="txt" href="radix_2.html#E3:26_2_1_1_1_1_2"><i><font color="Green" title="E41">A30</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E9:26_2_1_1_1_1_2"/>
<font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="binarith.html#T52" title="BINARITH:th.52">BINARITH:52</a></i>;<br/><a NAME="E10:26_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E46">A35</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="radix_2.html#E8:26_2_1_1_1_1_2"><i><font color="Green" title="E45">A34</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E11:26_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E47">A36</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">en</font>
 <b>by </b><i><a class="txt" href="radix_2.html#E6:26_2"><i><font color="Green" title="E24">A12</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E12:26_2_1_1_1_1_2"/><i><font color="Green" title="E48">A37</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T52" title="BINARITH:th.52">BINARITH:52</a></i>;<br/><a NAME="E13:26_2_1_1_1_1_2"/>
<font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="radix_2.html#E2:26_2_1_1"><i><font color="Green" title="E33">A21</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">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" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E15:26_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c10">nn</font>
 <b>by </b><i><a class="ref" href="binarith.html#T59" title="BINARITH:th.59">BINARITH:59</a></i>;<br/><a NAME="E16:26_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E17:26_2_1_1_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">i</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E18:26_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="radix_2.html#E12:26_2_1_1_1_1_2"><i><font color="Green" title="E48">A37</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E19:26_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E49">A38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">e</font>
 <b>by </b><i><a class="txt" href="radix_2.html#E8:26_2_1_1"><i><font color="Green" title="E38">A26</font></i></a>, <a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/><span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">en</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">i</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
<b>by </b><i><a class="txt" href="radix_2.html#E1:26_2_1_1"><i><font color="Green" title="E32">A20</font></i></a>, <a class="txt" href="radix_2.html#E3:26_2_1_1_1_1_2"><i><font color="Green" title="E41">A30</font></i></a>, <a class="txt" href="radix_2.html#E5:26_2_1_1_1_1_2"><i><font color="Green" title="E43">A32</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="txt" href="radix_2.html#E11:26_2_1_1_1_1_2"><i><font color="Green" title="E47">A36</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
<b>by </b><i><a class="txt" href="radix_2.html#E7:26_2"><i><font color="Green" title="E25">A13</font></i></a>, <a class="txt" href="radix_2.html#E10:26_2_1_1_1_1_2"><i><font color="Green" title="E46">A35</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"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">i</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
<b>by </b><i><a class="txt" href="radix_2.html#E3:26_2_1_1_1_1_2"><i><font color="Green" title="E41">A30</font></i></a>, <a class="ref" href="binarith.html#T56" title="BINARITH:th.56">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" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K8" title="RADIX_2:func.8">Table2</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">i</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
<b>by </b><i><a class="txt" href="radix_2.html#E19:26_2_1_1_1_1_2"><i><font color="Green" title="E49">A38</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E21:26_2_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="radix_2.html#E6:26_2_1_1_1_1_2"><i><font color="Green" title="E44">A33</font></i></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" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:26_2_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="radix_2.html#E1:26_2_1"><i><font color="Green" title="E30">A18</font></i></a>, <a class="txt" href="radix_2.html#E2:26_2_1"><i><font color="Green" title="E31">A19</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:26_2_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Olive" title="b1">i</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E14:26_2"/><i><font color="Green" title="E30">A39</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="radix_2.html#E1:26_2"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">ien</font> =  <a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font>;<br/>
<a NAME="E15:26_2"/><i><font color="Green" title="E31">A40</font></i>: 
<font color="Maroon" title="c5">ie</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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" title="c10">r2</font> =  <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  = <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> $1;<br/>
<b>consider </b><font color="Maroon" title="c11">rD</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E3:26_2_2"/><i><font color="Green" title="E32">A41</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">rD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>and </b><br/><a NAME="E4:26_2_2"/><i><font color="Green" title="E33">A42</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c11">rD</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">i</font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c12">rD</font> = <font color="Maroon" title="c11">rD</font> as   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c1">n</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="txt" href="radix_2.html#E3:26_2_2"><i><font color="Green" title="E32">A41</font></i></a>, <a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<i><font color="Green" title="E34">A43</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>
;<br/>
<a NAME="E7:26_2_2"/><i><font color="Green" title="E35">A44</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">rD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="radix_2.html#E3:26_2_2"><i><font color="Green" title="E32">A41</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:26_2_2"/><i><font color="Green" title="E36">A45</font></i>: 
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
;<br/>
<a NAME="E9:26_2_2"/><i><font color="Green" title="E37">A46</font></i>: 
 <a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">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" title="E38">A47</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<i><font color="Green" title="E39">A48</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">r2</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c12">rD</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_2.html#E4:26_2_2"><i><font color="Green" title="E33">A42</font></i></a>, <a class="txt" href="radix_2.html#E6:26_2_2"><i><font color="Green" title="E34">A43</font></i></a>, <a class="txt" href="radix_2.html#E7:26_2_2"><i><font color="Green" title="E35">A44</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>
;<br/>
<a NAME="E3:26_2_2_2"/><i><font color="Green" title="E40">A49</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/>
<i><font color="Green" title="E41">A50</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="txt" href="radix_2.html#E2:26_2_2_2"><i><font color="Green" title="E39">A48</font></i></a>, <a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>
;<br/>
<a NAME="E5:26_2_2_2"/><b>then </b><i><font color="Green" title="E42">A51</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E6:26_2_2_2"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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" title="c13">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:26_2_2_2_3"/><i><font color="Green" title="E43">A52</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">i</font> &amp; <font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font></span>)</span> )
 ;<br/>

<a NAME="E2:26_2_2_2_3"/><b>then </b><i><font color="Green" title="E44">A53</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E3:26_2_2_2_3"/><b>then </b><i><font color="Green" title="E45">A54</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="radix_2.html#E1:26_2_2_2_3"><i><font color="Green" title="E43">A52</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:26_2_2_2_3"/><b>then </b><i><font color="Green" title="E46">A55</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="radix_2.html#E1:26_2_2_2"><i><font color="Green" title="E38">A47</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c13">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&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" title="E47">A56</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> = 
 <a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T58" title="FINSEQ_1:th.58">FINSEQ_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
<b>by </b><i><a class="txt" href="radix_2.html#E10:26_2"><i><font color="Green" title="E26">A14</font></i></a>, <a class="ref" href="radix_2.html#D2" target="_self" title="RADIX_2:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> 1
<b>by </b><i><a class="txt" href="radix_2.html#E4:26_2_2_2_3"><i><font color="Green" title="E46">A55</font></i></a>, <a class="txt" href="radix_2.html#E1:26_2_2_2_3_1_1"><i><font color="Green" title="E47">A56</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font>
 <b>by </b><i><a class="txt" href="radix_2.html#E1:26_2_2_2_3_1_1"><i><font color="Green" title="E47">A56</font></i></a></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" title="E47">A57</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:26_2_2_2_3_1_2"/>
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="txt" href="radix_2.html#E1:26_2_2_2_3"><i><font color="Green" title="E43">A52</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c14">i1</font> = <font color="Maroon" title="c13">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E4:26_2_2_2_3_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">i</font>
 <b>by </b><i><a class="txt" href="radix_2.html#E1:26_2_2_2_3"><i><font color="Green" title="E43">A52</font></i></a>, <a class="txt" href="radix_2.html#E1:26_2_2_2_3_1_2"><i><font color="Green" title="E47">A57</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">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" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">i</font>
 <b>by </b><i><a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/><a NAME="E6:26_2_2_2_3_1_2"/><b>then </b><i><font color="Green" title="E48">A58</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E7:26_2_2_2_3_1_2"/><i><font color="Green" title="E49">A59</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="txt" href="radix_2.html#E2:26_2_2_2_3"><i><font color="Green" title="E44">A53</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E8:26_2_2_2_3_1_2"/><b>then </b><i><font color="Green" title="E50">A60</font></i>: 
<font color="Maroon" title="c14">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="radix_2.html#E6:26_2_2_2_3_1_2"><i><font color="Green" title="E48">A58</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E9:26_2_2_2_3_1_2"/><i><font color="Green" title="E51">A61</font></i>: 
<font color="Maroon" title="c14">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">rD</font>
 <b>by </b><i><a class="txt" href="radix_2.html#E7:26_2_2"><i><font color="Green" title="E35">A44</font></i></a>, <a class="txt" href="radix_2.html#E6:26_2_2_2_3_1_2"><i><font color="Green" title="E48">A58</font></i></a>, <a class="txt" href="radix_2.html#E7:26_2_2_2_3_1_2"><i><font color="Green" title="E49">A59</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c15">rs</font> = <font color="Maroon" title="c12">rD</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">rs2</font> = <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="radix_2.html#E6:26_2_2"><i><font color="Green" title="E34">A43</font></i></a>, <a class="txt" href="radix_2.html#E7:26_2_2"><i><font color="Green" title="E35">A44</font></i></a>, <a class="txt" href="radix_2.html#E9:26_2_2_2_3_1_2"><i><font color="Green" title="E51">A61</font></i></a>, <a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/><a NAME="E12:26_2_2_2_3_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">i</font>
 <b>by </b><i><a class="txt" href="radix_2.html#E1:26_2_2_2_3"><i><font color="Green" title="E43">A52</font></i></a>, <a class="txt" href="radix_2.html#E1:26_2_2_2_3_1_2"><i><font color="Green" title="E47">A57</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="radix_2.html#E3:26_2_2_2"><i><font color="Green" title="E40">A49</font></i></a>, <a class="txt" href="radix_2.html#E4:26_2_2_2"><i><font color="Green" title="E41">A50</font></i></a>, <a class="txt" href="radix_2.html#E2:26_2_2_2_3"><i><font color="Green" title="E44">A53</font></i></a>, <a class="ref" href="finseq_1.html#T37" title="FINSEQ_1:th.37">FINSEQ_1:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">r2</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c12">rD</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="radix_2.html#E4:26_2_2"><i><font color="Green" title="E33">A42</font></i></a>, <a class="txt" href="radix_2.html#E6:26_2_2"><i><font color="Green" title="E34">A43</font></i></a>, <a class="txt" href="radix_2.html#E7:26_2_2"><i><font color="Green" title="E35">A44</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>
<br/>.= 
<font color="Maroon" title="c10">r2</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c15">rs</font>
<b>by </b><i><a class="txt" href="radix_2.html#E8:26_2_2_2_3_1_2"><i><font color="Green" title="E50">A60</font></i></a>, <a class="ref" href="rvsum_1.html#T67" title="RVSUM_1:th.67">RVSUM_1:67</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c16">rs2</font>
<b>by </b><i><a class="txt" href="radix_2.html#E4:26_2_2"><i><font color="Green" title="E33">A42</font></i></a>, <a class="txt" href="radix_2.html#E6:26_2_2"><i><font color="Green" title="E34">A43</font></i></a>, <a class="txt" href="radix_2.html#E7:26_2_2"><i><font color="Green" title="E35">A44</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c14">i1</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_2.html#E6:26_2_2"><i><font color="Green" title="E34">A43</font></i></a>, <a class="txt" href="radix_2.html#E7:26_2_2"><i><font color="Green" title="E35">A44</font></i></a>, <a class="txt" href="radix_2.html#E9:26_2_2_2_3_1_2"><i><font color="Green" title="E51">A61</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c14">i1</font>,<font color="Maroon" title="c3">k</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_2.html#E8:26_2_2_2_3_1_2"><i><font color="Green" title="E50">A60</font></i></a>, <a class="ref" href="radix_2.html#D2" target="_self" title="RADIX_2:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">en</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c14">i1</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">en</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c14">i1</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c14">i1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">en</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c14">i1</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_2.html#E6:26_2_2_2_3_1_2"><i><font color="Green" title="E48">A58</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c14">i1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="radix_2.html#E7:26_2"><i><font color="Green" title="E25">A13</font></i></a>, <a class="txt" href="radix_2.html#E8:26_2_2_2_3_1_2"><i><font color="Green" title="E50">A60</font></i></a></i>
<br/>.= 
 <a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c3">k</font>
<b>by </b><i><a class="txt" href="radix_2.html#E1:26_2_2_2_3"><i><font color="Green" title="E43">A52</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c13">i</font>
<b>by </b><i><a class="txt" href="radix_2.html#E3:26_2_2_2_3"><i><font color="Green" title="E45">A54</font></i></a>, <a class="ref" href="radix_2.html#D2" target="_self" title="RADIX_2:def.2">Def2</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" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font>
 <b>by </b><i><a class="txt" href="radix_2.html#E4:26_2_2_2_3"><i><font color="Green" title="E46">A55</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="radix_2.html#E5:26_2_2_2"><i><font color="Green" title="E42">A51</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:26_2_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">e</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E11:26_2_2"/><b>then </b><i><font color="Green" title="E38">A62</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">e</font>
 
<b>by </b><i><a class="txt" href="radix_2.html#E10:26_2"><i><font color="Green" title="E26">A14</font></i></a>, <a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<font color="Maroon" title="c5">ie</font> = 
 <a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">k</font>
<b>by </b><i><a class="txt" href="radix_2.html#E2:26_2"><i><font color="Green" title="E21">A9</font></i></a>, <a class="txt" href="radix_2.html#E3:26_2"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="radix_2.html#E4:26_2"><i><font color="Green" title="E23">A11</font></i></a>, <a class="ref" href="radix_2.html#T15" target="_self" title="RADIX_2:th.15">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="radix_2.html#E9:26_2_2"><i><font color="Green" title="E37">A46</font></i></a>, <a class="ref" href="rvsum_1.html#T106" title="RVSUM_1:th.106">RVSUM_1:106</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">r2</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c12">rD</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="radix_2.html#E4:26_2_2"><i><font color="Green" title="E33">A42</font></i></a>, <a class="txt" href="radix_2.html#E6:26_2_2"><i><font color="Green" title="E34">A43</font></i></a>, <a class="txt" href="radix_2.html#E7:26_2_2"><i><font color="Green" title="E35">A44</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">r2</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c12">rD</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T117" title="RVSUM_1:th.117">RVSUM_1:117</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c6">e</font>,1,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="radix_2.html#E4:26_2_2"><i><font color="Green" title="E33">A42</font></i></a>, <a class="txt" href="radix_2.html#E6:26_2_2"><i><font color="Green" title="E34">A43</font></i></a>, <a class="txt" href="radix_2.html#E7:26_2_2"><i><font color="Green" title="E35">A44</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">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" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 0</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="radix_2.html#E8:26_2_2"><i><font color="Green" title="E36">A45</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>
;<br/>
<b>hence </b><a NAME="E13:26_2_2"/>
<font color="Maroon" title="c5">ie</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="radix_2.html#E11:26_2_2"><i><font color="Green" title="E38">A62</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:26_2"/><b>then </b><i><font color="Green" title="E32">A63</font></i>: 
<font color="Maroon" title="c5">ie</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="int_1.html#T19" title="INT_1:th.19">INT_1:19</a></i>;<br/>
<a NAME="E17:26_2"/><i><font color="Green" title="E33">A64</font></i>: 
 <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="radix_2.html#T6" target="_self" title="RADIX_2:th.6">Th6</a></i>;<br/>
<a NAME="E18:26_2"/>
<font color="Maroon" title="c5">ie</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="radix_2.html#E3:26_2"><i><font color="Green" title="E22">A10</font></i></a>, <a class="ref" href="radix_1.html#D12" title="RADIX_1:def.12">RADIX_1:def 12</a></i>;<br/>
<a NAME="E19:26_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="radix_2.html#E16:26_2"><i><font color="Green" title="E32">A63</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E20:26_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>;<br/>
<a NAME="E21:26_2"/><b>then </b>
 <a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<a NAME="E22:26_2"/><b>then </b><i><font color="Green" title="E34">A65</font></i>: 
 <a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">k</font>
 
<b>by </b><i><a class="ref" href="radix_1.html#D12" title="RADIX_1:def.12">RADIX_1:def 12</a></i>;<br/>
<a NAME="E23:26_2"/><i><font color="Green" title="E35">A66</font></i>: 
<font color="Maroon" title="c8">en</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <span class="p1">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">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" title="E36">A67</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">en</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <span class="p2">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="radix_2.html#E6:26_2"><i><font color="Green" title="E24">A12</font></i></a>, <a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E2:26_2_3"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">en</font> holds <br/><font color="Maroon" title="c8">en</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <span class="p2">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Olive" title="b1">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" title="c10">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:26_2_3_1"/><i><font color="Green" title="E37">A68</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">en</font> )
 ;<br/>

<a NAME="E2:26_2_3_1"/><b>then </b><i><font color="Green" title="E38">A69</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="radix_2.html#E6:26_2"><i><font color="Green" title="E24">A12</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:26_2_3_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="radix_2.html#E6:26_2"><i><font color="Green" title="E24">A12</font></i></a>, <a class="txt" href="radix_2.html#E1:26_2_3_1"><i><font color="Green" title="E37">A68</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E4:26_2_3_1"/><b>then </b><i><font color="Green" title="E39">A70</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<font color="Maroon" title="c6">e</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> 1 = 
 <a href="radix_2.html#K4" title="RADIX_2:func.4">DigitDC2</a> <font color="Maroon" title="c5">ie</font>,1,<font color="Maroon" title="c3">k</font>
<b>by </b><i><a class="txt" href="radix_2.html#E4:26_2"><i><font color="Green" title="E23">A11</font></i></a>, <a class="txt" href="radix_2.html#E10:26_2"><i><font color="Green" title="E26">A14</font></i></a>, <a class="ref" href="radix_2.html#D5" target="_self" title="RADIX_2:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">ie</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">ie</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 1
<b>by </b><i><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>
<br/>.= 
<font color="Maroon" title="c5">ie</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="nat_2.html#T6" title="NAT_2:th.6">NAT_2:6</a></i>
<br/>.= 
<font color="Maroon" title="c5">ie</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></i>
;<br/>
<a NAME="E6:26_2_3_1"/><b>then </b><i><font color="Green" title="E40">A71</font></i>: 
<font color="Maroon" title="c6">e</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font>
 
<b>by </b><i><a class="txt" href="radix_2.html#E17:26_2"><i><font color="Green" title="E33">A64</font></i></a>, <a class="ref" href="nat_d.html#T1" title="NAT_D:th.1">NAT_D:1</a></i>;<br/>
<a NAME="E7:26_2_3_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">e</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:26_2_3_1"/><b>then </b><i><font color="Green" title="E41">A72</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E9:26_2_3_1"/><i><font color="Green" title="E42">A73</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font> )
 
<b>by </b><i><a class="txt" href="radix_2.html#E1:26_2_3_1"><i><font color="Green" title="E37">A68</font></i></a>, <a class="ref" href="radix_2.html#T6" target="_self" title="RADIX_2:th.6">Th6</a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E10:26_2_3_1"/><i><font color="Green" title="E43">A74</font></i>: 
(  <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="radix_2.html#E1:26_2_3_1"><i><font color="Green" title="E37">A68</font></i></a>, <a class="ref" href="radix_2.html#T6" target="_self" title="RADIX_2:th.6">Th6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">a</font> = <font color="Maroon" title="c6">e</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> 1 as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c12">b</font> =  <a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c13">G</font> = <font color="Maroon" title="c11">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<i><font color="Green" title="E44">A75</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> = 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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" title="c6">e</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p5">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<b>by </b><i><a class="ref" href="int_1.html#D7" title="INT_1:def.7">INT_1:def 7</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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" title="c6">e</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<b>by </b><i><a class="txt" href="radix_2.html#E9:26_2_3_1"><i><font color="Green" title="E42">A73</font></i></a>, <a class="ref" href="xcmplx_1.html#T114" title="XCMPLX_1:th.114">XCMPLX_1:114</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span>
<b>by </b><i><a class="ref" href="int_1.html#T51" title="INT_1:th.51">INT_1:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span>
<b>by </b><i><a class="ref" href="int_1.html#D7" title="INT_1:def.7">INT_1:def 7</a></i>
<br/>.= 
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_2.html#E6:26_2_3_1"><i><font color="Green" title="E40">A71</font></i></a>, <a class="ref" href="nat_d.html#T27" title="NAT_D:th.27">NAT_D:27</a></i>
;<br/>
<font color="Maroon" title="c8">en</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">i</font> = 
<font color="Maroon" title="c6">e</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="radix_2.html#E7:26_2"><i><font color="Green" title="E25">A13</font></i></a>, <a class="txt" href="radix_2.html#E2:26_2_3_1"><i><font color="Green" title="E38">A69</font></i></a></i>
<br/>.= 
 <a href="radix_2.html#K4" title="RADIX_2:func.4">DigitDC2</a> <font color="Maroon" title="c5">ie</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">k</font>
<b>by </b><i><a class="txt" href="radix_2.html#E4:26_2"><i><font color="Green" title="E23">A11</font></i></a>, <a class="txt" href="radix_2.html#E4:26_2_3_1"><i><font color="Green" title="E39">A70</font></i></a>, <a class="ref" href="radix_2.html#D5" target="_self" title="RADIX_2:def.5">Def5</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" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_2.html#E15:26_2"><i><font color="Green" title="E31">A40</font></i></a>, <a class="txt" href="radix_2.html#E9:26_2_3_1"><i><font color="Green" title="E42">A73</font></i></a>, <a class="ref" href="radix_2.html#T4" target="_self" title="RADIX_2:th.4">Th4</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" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_2.html#E10:26_2"><i><font color="Green" title="E26">A14</font></i></a>, <a class="txt" href="radix_2.html#E8:26_2_3_1"><i><font color="Green" title="E41">A72</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c10">i</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">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" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_2.html#E10:26_2_3_1"><i><font color="Green" title="E43">A74</font></i></a>, <a class="ref" href="pepin.html#T27" title="PEPIN:th.27">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" title="c6">e</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p5">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span>
<b>by </b><i><a class="ref" href="nat_2.html#T29" title="NAT_2:th.29">NAT_2:29</a></i>
<br/>.= 
 <a href="radix_2.html#K4" title="RADIX_2:func.4">DigitDC2</a> <span class="p1">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span>,<font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c3">k</font>
<b>by </b><i><a class="txt" href="radix_2.html#E1:26_2_3_1"><i><font color="Green" title="E37">A68</font></i></a>, <a class="txt" href="radix_2.html#E9:26_2_3_1"><i><font color="Green" title="E42">A73</font></i></a>, <a class="txt" href="radix_2.html#E14:26_2_3_1"><i><font color="Green" title="E44">A75</font></i></a>, <a class="ref" href="radix_2.html#T4" target="_self" title="RADIX_2:th.4">Th4</a></i>
;<br/>
<b>hence </b><a NAME="E16:26_2_3_1"/>
<font color="Maroon" title="c8">en</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <span class="p2">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="txt" href="radix_2.html#E2:26_2_3_1"><i><font color="Green" title="E38">A69</font></i></a>, <a class="ref" href="radix_2.html#D5" target="_self" title="RADIX_2:def.5">Def5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:26_2_3"/>
<font color="Maroon" title="c8">en</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <span class="p1">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="radix_2.html#E1:26_2_3"><i><font color="Green" title="E36">A67</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E24:26_2"/><i><font color="Green" title="E36">A76</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
;<br/>
<a NAME="E25:26_2"/>
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E26:26_2"/><b>then </b>
 ex <font color="Olive" title="b1">I1</font>, <font color="Olive" title="b2">I2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">I1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b2">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">I1</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K8" title="RADIX_2:func.8">Table2</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font> )
 
<b>by </b><i><a class="txt" href="radix_2.html#E1:26_2"><i><font color="Green" title="E20">A8</font></i></a>, <a class="txt" href="radix_2.html#E24:26_2"><i><font color="Green" title="E36">A76</font></i></a>, <a class="ref" href="radix_2.html#D9" target="_self" title="RADIX_2:def.9">Def9</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K8" title="RADIX_2:func.8">Table2</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
<b>by </b><i><a class="txt" href="radix_2.html#E13:26_2"><i><font color="Green" title="E29">A17</font></i></a>, <a class="txt" href="radix_2.html#E14:26_2"><i><font color="Green" title="E30">A39</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" title="c2">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p0">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K8" title="RADIX_2:func.8">Table2</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
<b>by </b><i><a class="txt" href="radix_2.html#E1:26_2"><i><font color="Green" title="E20">A8</font></i></a>, <a class="txt" href="radix_2.html#E3:26_2"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="radix_2.html#E22:26_2"><i><font color="Green" title="E34">A65</font></i></a>, <a class="txt" href="radix_2.html#E23:26_2"><i><font color="Green" title="E35">A66</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"><font color="Maroon" title="c2">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p5">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K8" title="RADIX_2:func.8">Table2</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
<b>by </b><i><a class="ref" href="pepin.html#T12" title="PEPIN:th.12">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" title="c2">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K8" title="RADIX_2:func.8">Table2</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
<b>by </b><i><a class="ref" href="euler_2.html#T10" title="EULER_2:th.10">EULER_2:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K8" title="RADIX_2:func.8">Table2</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
<b>by </b><i><a class="ref" href="newton.html#T14" title="NEWTON:th.14">NEWTON:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
<b>by </b><i><a class="ref" href="euler_2.html#T9" title="EULER_2:th.9">EULER_2:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</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" title="RADIX_2:func.3">SDDec2</a> <font color="Maroon" title="c8">en</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
<b>by </b><i><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></i>
;<br/>
<b>hence </b><a NAME="E28:26_2"/>
<span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c5">ie</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="radix_2.html#E15:26_2"><i><font color="Green" title="E31">A40</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:26"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S8" title="NAT_1:sch.8">NAT_1:sch 8</a>(<a class="txt" href="radix_2.html#E1:26"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="radix_2.html#E2:26"><i><font color="Green" title="E19">A7</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:26"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/>for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k</font>, <font color="Olive" title="b4">f</font>, <font color="Olive" title="b5">ie</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b5">ie</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> <font color="Olive" title="b1">n</font>,<font color="Olive" title="b3">k</font> &amp; <font color="Olive" title="b4">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 holds <br/>for <font color="Olive" title="b6">e</font> being  <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Olive" title="b1">n</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> <font color="Olive" title="b5">ie</font>,<font color="Olive" title="b1">n</font>,<font color="Olive" title="b3">k</font> holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K9" title="RADIX_2:func.9">Pow_mod</a> <font color="Olive" title="b2">m</font>,<font color="Olive" title="b6">e</font>,<font color="Olive" title="b4">f</font>,<font color="Olive" title="b3">k</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Olive" title="b5">ie</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Olive" title="b4">f</font>
 ;<br/>


</div>
