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<b>let </b><font color="Maroon">f</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <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="E1:15"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp; <font color="Maroon">f</font> <a href="radix_6.html#R1" target="_self">needs_digits_of</a> <font color="Maroon">m</font>,<font color="Maroon">k</font> )
 ;<br/>

<a NAME="E2:15"/>
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/> <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">f</font>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6">Fmin</a> <font color="Maroon">m</font>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Olive">i</font>
 
<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">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_1"/><i><font color="Green">E25</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="E2:15_1"/><b>then </b><i><font color="Green">E26</font></i>: 
( <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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> or <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_1_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><a NAME="E2:15_1_1_1_1"/><b>then </b>
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">f</font>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, </i>;<br/><a NAME="E3:15_1_1_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">f</font>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/> <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6">Fmin</a> <font color="Maroon">m</font>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> = 
 <a href="radix_5.html#K5">FminDigit</a> <font color="Maroon">m</font>,<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/>.= 
1
<b>by </b><i>, <a class="txt" href="radix_6.html#E1"><i><font color="Green">Lemma18</font></i></a>, <a class="ref" href="radix_5.html#D5">RADIX_5:def 5</a></i>
;<br/><b>hence </b><a NAME="E5:15_1_1_1_1"/>
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">f</font>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6">Fmin</a> <font color="Maroon">m</font>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_1_1_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"> <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6">Fmin</a> <font color="Maroon">m</font>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> = 
 <a href="radix_5.html#K5">FminDigit</a> <font color="Maroon">m</font>,<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="txt" href="radix_6.html#E1"><i><font color="Green">Lemma18</font></i></a>, <a class="ref" href="radix_5.html#D5">RADIX_5:def 5</a></i>
;<br/><b>hence </b><a NAME="E3:15_1_1_1_2"/>
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">f</font>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6">Fmin</a> <font color="Maroon">m</font>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:15_1"/>
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">f</font>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><a href="radix_5.html#K6">Fmin</a> <font color="Maroon">m</font>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>,<font color="Maroon">i</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:15"/><b>then </b>
 <a href="radix_1.html#K8">SDDec</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">f</font>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <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">m</font>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="radix_5.html#T13">RADIX_5:13</a></i>;<br/>
<a NAME="E4:15"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="radix_1.html#K8">SDDec</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">f</font>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <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> 2</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:15"/>
<font color="Maroon">f</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">m</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:15"/><b>then </b>
<font color="Maroon">f</font> <a href="radix_1.html#R1">is_represented_by</a> <font color="Maroon">m</font>,<font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="radix_1.html#D12">RADIX_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E7:15"/>
<font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <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> 2</span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="radix_1.html#T25">RADIX_1:25</a></i>;<br/>


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