<?xml version="1.0"?>
<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>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 holds <br/> <a href="radix_1.html#K8">SDDec</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6">Fmin</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>1</sub></font>,<font color="Olive">k</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0;<br/>
<a NAME="E1:3"/><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="3_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>assume </b><a NAME="E1:3_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 ;<br/>

<a NAME="E2:3_1"/><i><font color="Green">E26</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/>
<b>then </b><span class="p1">(<span class="default"><a href="radix_1.html#K7">DigitSD</a> <span class="p2">(<span class="default"><a href="radix_5.html#K6">Fmin</a> 1,1,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
 <a href="radix_1.html#K6">SubDigit</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6">Fmin</a> 1,1,<font color="Maroon">k</font></span>)</span>,1,<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="radix_1.html#D6">RADIX_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K5">DigB</a> <span class="p2">(<span class="default"><a href="radix_5.html#K6">Fmin</a> 1,1,<font color="Maroon">k</font></span>)</span>,1</span>)</span>
<b>by </b><i><a class="ref" href="radix_1.html#D5">RADIX_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p2">(<span class="default"><a href="radix_5.html#K6">Fmin</a> 1,1,<font color="Maroon">k</font></span>)</span>,1</span>)</span>
<b>by </b><i><a class="ref" href="radix_1.html#D4">RADIX_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> 0</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p2">(<span class="default"><a href="radix_5.html#K6">Fmin</a> 1,1,<font color="Maroon">k</font></span>)</span>,1</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p2">(<span class="default"><a href="radix_5.html#K6">Fmin</a> 1,1,<font color="Maroon">k</font></span>)</span>,1</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>
<br/>.= 
 <a href="radix_5.html#K5">FminDigit</a> 1,<font color="Maroon">k</font>,1
<b>by </b><i>, <a class="ref" href="radix_5.html#D6">RADIX_5:def 6</a></i>
<br/>.= 
1
<b>by </b><i>, <a class="ref" href="radix_5.html#D5">RADIX_5:def 5</a></i>
;<br/>
<a NAME="E4:3_1"/><b>then </b><i><font color="Green">E27</font></i>: 
 <a href="radix_1.html#K7">DigitSD</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6">Fmin</a> 1,1,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="radix_1.html#T20">RADIX_1:20</a></i>;<br/>
 <a href="radix_1.html#K8">SDDec</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6">Fmin</a> 1,1,<font color="Maroon">k</font></span>)</span> = 
 <a href="wsierp_1.html#K6">Sum</a> <span class="p1">(<span class="default"><a href="radix_1.html#K7">DigitSD</a> <span class="p2">(<span class="default"><a href="radix_5.html#K6">Fmin</a> 1,1,<font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="radix_1.html#D7">RADIX_1:def 7</a></i>
<br/>.= 
1
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T103">RVSUM_1:103</a></i>
;<br/>
<b>hence </b><a NAME="E6:3_1"/>
 <a href="radix_1.html#K8">SDDec</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6">Fmin</a> 1,1,<font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:3"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:3_2"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</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>assume </b><a NAME="E2:3_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 ;<br/>

<a NAME="E3:3_2"/>
<font color="Maroon">m</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">m</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/>
<b>then </b><i><font color="Green">E32</font></i>:  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="radix_5.html#K5">FminDigit</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="radix_5.html#D6">RADIX_5:def 6</a></i>
<br/>.= 
1
<b>by </b><i>, <a class="ref" href="radix_5.html#D5">RADIX_5:def 5</a></i>
;<br/>
<a NAME="E5:3_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">m</font> holds <br/><span class="p1">(<span class="default"><a href="radix_5.html#K6">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10">DecSD</a> 0,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_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="E1:3_2_2"/><i><font color="Green">E33</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">m</font>
 ;<br/>

<a NAME="E2:3_2_2"/><b>then </b><i><font color="Green">E34</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">m</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="E3:3_2_2"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:3_2_2"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<span class="p1">(<span class="default"><a href="radix_5.html#K6">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span>,<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_5.html#K5">FminDigit</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font>,<font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="radix_5.html#D6">RADIX_5:def 6</a></i>
<br/>.= 
0
<b>by </b><i>, , <a class="ref" href="radix_5.html#D5">RADIX_5:def 5</a></i>
<br/>.= 
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10">DecSD</a> 0,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="radix_5.html#T5">RADIX_5:5</a></i>
;<br/>
<b>hence </b><a NAME="E6:3_2_2"/>
<span class="p1">(<span class="default"><a href="radix_5.html#K6">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10">DecSD</a> 0,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <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>then </b><i><font color="Green">E36</font></i>:  <a href="radix_1.html#K8">SDDec</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> 0,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span></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">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p3">(<span class="default"><a href="radix_5.html#K6">Fmin</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="radix_2.html#T10">RADIX_2:10</a></i>
<br/>.= 
0 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="radix_5.html#T6">RADIX_5:6</a></i>
;<br/>
<a NAME="E7:3_2"/>
 <a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2
 
<b>by </b><i>, <a class="ref" href="radix_4.html#T1">RADIX_4:1</a></i>;<br/>
<a NAME="E8:3_2"/><b>then </b>
 <a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:3_2"/><b>then </b>
 <a href="radix_1.html#K8">SDDec</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i>, <a class="ref" href="prepower.html#T19">PREPOWER:19</a></i>;<br/>
<b>hence </b><a NAME="E10:3_2"/>
 <a href="radix_1.html#K8">SDDec</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6">Fmin</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:3"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</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">m</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="E4:3"/>
for <font color="Olive">m</font>, <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 holds <br/> <a href="radix_1.html#K8">SDDec</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6">Fmin</a> <font color="Olive">m</font>,<font color="Olive">m</font>,<font color="Olive">k</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>


</div>
