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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">tx</font>, <font color="Olive">ty</font> being  <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">a<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Olive">k</font> <a href="radix_1.html#K3">-SD</a> </span>)</span>  st ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="radix_1.html#K4">DigA</a> <font color="Olive">tx</font>,<font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Olive">ty</font>,<font color="Olive">i</font> ) holds <br/> <a href="radix_1.html#K8">SDDec</a> <font color="Olive">tx</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="radix_1.html#K8">SDDec</a> <font color="Olive">ty</font>;<br/>
<a NAME="E1:15"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">tx</font> be   <a href="binarith.html#NM1">Tuple</a> of 1,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="radix_1.html#K3">-SD</a> </span>)</span>, <font color="Maroon">ty</font> be   <a href="binarith.html#NM1">Tuple</a> of 1,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="radix_1.html#K3">-SD</a> </span>)</span>;<br/>



<b>assume </b><a NAME="E1:15_1"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1 holds <br/> <a href="radix_1.html#K4">DigA</a> <font color="Maroon">tx</font>,<font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">ty</font>,<font color="Olive">i</font>
 ;<br/>

<a NAME="E2:15_1"/><i><font color="Green">E23</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:15_1"/><i><font color="Green">E30</font></i>: 
 <a href="radix_1.html#K8">SDDec</a> <font color="Maroon">tx</font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">tx</font>,1
 
<b>by </b><i/>;<br/>
<a NAME="E4:15_1"/>
 <a href="radix_1.html#K8">SDDec</a> <font color="Maroon">ty</font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">ty</font>,1
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:15_1"/>
 <a href="radix_1.html#K8">SDDec</a> <font color="Maroon">tx</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="radix_1.html#K8">SDDec</a> <font color="Maroon">ty</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:15"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:15_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>] )
 ;<br/>

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">tx</font> be   <a href="binarith.html#NM1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="radix_1.html#K3">-SD</a> </span>)</span>, <font color="Maroon">ty</font> be   <a href="binarith.html#NM1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="radix_1.html#K3">-SD</a> </span>)</span>;<br/>



<b>assume </b><a NAME="E2:15_2"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/> <a href="radix_1.html#K4">DigA</a> <font color="Maroon">tx</font>,<font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">ty</font>,<font color="Olive">i</font>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="mesfunc1.html#K2">INT</a>  =  <a href="radix_1.html#K5">DigB</a> <font color="Maroon">tx</font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">txn</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="mesfunc1.html#K2">INT</a> <b> such that </b><br/><a NAME="E4:15_2"/><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">txn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E5:15_2"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">txn</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<a NAME="E6:15_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">txn</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">k</font> <a href="radix_1.html#K3">-SD</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:15_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">txn</font>
 ;<br/>

<b>then consider </b><font color="Maroon">xx</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:15_2_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">xx</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">txn</font>
 <b>and </b><br/><a NAME="E4:15_2_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">txn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">xx</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">xx</font> = <font color="Maroon">xx</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E6:15_2_1"/><i><font color="Green">E53</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">txn</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then </b><i><font color="Green">E54</font></i>: <font color="Maroon">z</font> = 
 <a href="radix_1.html#K5">DigB</a> <font color="Maroon">tx</font>,<font color="Maroon">xx</font>
<b>by </b><i><a class="ref" href="radix_5.html#T1" target="_self">Th1</a>, , <a class="txt" href="radix_5.html#E1:15"><i><font color="Green">E20</font></i></a></i>
<br/>.= 
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">tx</font>,<font color="Maroon">xx</font>
<b>by </b><i><a class="ref" href="radix_1.html#D4">RADIX_1:def 4</a></i>
;<br/>
<a NAME="E8:15_2_1"/>
<font color="Maroon">xx</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_2.html#T9">FINSEQ_2:9</a></i>;<br/>
<a NAME="E9:15_2_1"/><b>then </b>
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">tx</font>,<font color="Maroon">xx</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">k</font> <a href="radix_1.html#K3">-SD</a> 
 
<b>by </b><i><a class="ref" href="radix_1.html#T19">RADIX_1:19</a></i>;<br/>
<b>hence </b><a NAME="E10:15_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="radix_1.html#K3">-SD</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">txn</font> = <font color="Maroon">txn</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">k</font> <a href="radix_1.html#K3">-SD</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E8:15_2"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">txn</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">tx</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:15_2_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:15_2_2"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T9">FINSEQ_2:9</a></i>;<br/>
<font color="Maroon">txn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
 <a href="radix_1.html#K5">DigB</a> <font color="Maroon">tx</font>,<font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="radix_5.html#T1" target="_self">Th1</a>, <a class="txt" href="radix_5.html#E1:15"><i><font color="Green">E20</font></i></a></i>
<br/>.= 
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">tx</font>,<font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="radix_1.html#D4">RADIX_1:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E4:15_2_2"/>
<font color="Maroon">txn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">tx</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="radix_1.html#D3">RADIX_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="mesfunc1.html#K2">INT</a>  =  <a href="radix_1.html#K5">DigB</a> <font color="Maroon">ty</font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">tyn</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="mesfunc1.html#K2">INT</a> <b> such that </b><br/><a NAME="E10:15_2"/><i><font color="Green">E59</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">tyn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E11:15_2"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">tyn</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">i</font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<a NAME="E12:15_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">tyn</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">k</font> <a href="radix_1.html#K3">-SD</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:15_2_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">tyn</font>
 ;<br/>

<b>then consider </b><font color="Maroon">yy</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:15_2_3"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">yy</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">tyn</font>
 <b>and </b><br/><a NAME="E4:15_2_3"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">tyn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">yy</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">yy</font> = <font color="Maroon">yy</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E6:15_2_3"/><i><font color="Green">E64</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">tyn</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="radix_5.html#E1:15"><i><font color="Green">E20</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then </b><i><font color="Green">E65</font></i>: <font color="Maroon">z</font> = 
 <a href="radix_1.html#K5">DigB</a> <font color="Maroon">ty</font>,<font color="Maroon">yy</font>
<b>by </b><i>, , </i>
<br/>.= 
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">ty</font>,<font color="Maroon">yy</font>
<b>by </b><i><a class="ref" href="radix_1.html#D4">RADIX_1:def 4</a></i>
;<br/>
<a NAME="E8:15_2_3"/>
<font color="Maroon">yy</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="radix_5.html#T2" target="_self">Th2</a>, <a class="ref" href="finseq_2.html#T9">FINSEQ_2:9</a></i>;<br/>
<a NAME="E9:15_2_3"/><b>then </b>
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">ty</font>,<font color="Maroon">yy</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">k</font> <a href="radix_1.html#K3">-SD</a> 
 
<b>by </b><i><a class="ref" href="radix_1.html#T19">RADIX_1:19</a></i>;<br/>
<b>hence </b><a NAME="E10:15_2_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="radix_1.html#K3">-SD</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">tyn</font> = <font color="Maroon">tyn</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">k</font> <a href="radix_1.html#K3">-SD</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E14:15_2"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">tyn</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ty</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:15_2_4"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:15_2_4"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T9">FINSEQ_2:9</a></i>;<br/>
<font color="Maroon">tyn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
 <a href="radix_1.html#K5">DigB</a> <font color="Maroon">ty</font>,<font color="Maroon">i</font>
<b>by </b><i>, </i>
<br/>.= 
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">ty</font>,<font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="radix_1.html#D4">RADIX_1:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E4:15_2_4"/>
<font color="Maroon">tyn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ty</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="radix_5.html#T2" target="_self">Th2</a>, <a class="ref" href="radix_1.html#D3">RADIX_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">txn</font> = <font color="Maroon">txn</font> as   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="radix_1.html#K3">-SD</a> </span>)</span> <b>by </b><i>, <a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<b>reconsider </b><font color="Maroon">tyn</font> = <font color="Maroon">tyn</font> as   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="radix_1.html#K3">-SD</a> </span>)</span> <b>by </b><i><a class="txt" href="radix_5.html#E1:15"><i><font color="Green">E20</font></i></a>, <a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E17:15_2"/><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">txn</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">tx</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K8">SDDec</a> <font color="Maroon">tx</font>
 
<b>by </b><i>, <a class="ref" href="radix_2.html#T10">RADIX_2:10</a></i>;<br/>
<a NAME="E18:15_2"/><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">tyn</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">ty</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K8">SDDec</a> <font color="Maroon">ty</font>
 
<b>by </b><i>, <a class="ref" href="radix_2.html#T10">RADIX_2:10</a></i>;<br/>
<a NAME="E19:15_2"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/> <a href="radix_1.html#K4">DigA</a> <font color="Maroon">txn</font>,<font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">tyn</font>,<font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:15_2_5"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/>

<b>then </b><i><font color="Green">E72</font></i>:  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">txn</font>,<font color="Maroon">i</font> = 
<font color="Maroon">txn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="radix_1.html#D3">RADIX_1:def 3</a></i>
<br/>.= 
 <a href="radix_1.html#K5">DigB</a> <font color="Maroon">tx</font>,<font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="radix_5.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">tx</font>,<font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="radix_1.html#D4">RADIX_1:def 4</a></i>
;<br/>
<i><font color="Green">E73</font></i>:  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">tyn</font>,<font color="Maroon">i</font> = 
<font color="Maroon">tyn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="radix_1.html#D3">RADIX_1:def 3</a></i>
<br/>.= 
 <a href="radix_1.html#K5">DigB</a> <font color="Maroon">ty</font>,<font color="Maroon">i</font>
<b>by </b><i>, </i>
<br/>.= 
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">ty</font>,<font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="radix_1.html#D4">RADIX_1:def 4</a></i>
;<br/>
<a NAME="E4:15_2_5"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T9">FINSEQ_2:9</a></i>;<br/>
<b>hence </b><a NAME="E5:15_2_5"/>
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">txn</font>,<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">tyn</font>,<font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="radix_5.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:15_2"/><b>then </b><i><font color="Green">E74</font></i>: 
 <a href="radix_1.html#K8">SDDec</a> <font color="Maroon">txn</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="radix_1.html#K8">SDDec</a> <font color="Maroon">tyn</font>
 
<b>by </b><i/>;<br/>
<a NAME="E21:15_2"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E22:15_2"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">tx</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">ty</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E23:15_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">tx</font>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">ty</font>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<b>hence </b><a NAME="E24:15_2"/>
 <a href="radix_1.html#K8">SDDec</a> <font color="Maroon">tx</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="radix_1.html#K8">SDDec</a> <font color="Maroon">ty</font>
 <b>by </b><i>, <a class="ref" href="radix_5.html#T2" target="_self">Th2</a>, , <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:15"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S9">NAT_1:sch 9</a>(, );<br/></i>
<b>hence </b><a NAME="E4:15"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 holds <br/>for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">tx</font>, <font color="Olive">ty</font> being  <a href="binarith.html#NM1">Tuple</a> of <font color="Olive">n</font>,<span class="p1">(<span class="default"><font color="Olive">k</font> <a href="radix_1.html#K3">-SD</a> </span>)</span>  st ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Olive">n</font> holds <br/> <a href="radix_1.html#K4">DigA</a> <font color="Olive">tx</font>,<font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Olive">ty</font>,<font color="Olive">i</font> ) holds <br/> <a href="radix_1.html#K8">SDDec</a> <font color="Olive">tx</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="radix_1.html#K8">SDDec</a> <font color="Olive">ty</font>
 ;<br/>


</div>
