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<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">k</font>, <font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:5"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon" title="c1">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="c3">n</font>
 ;<br/>

<a NAME="E2:5"/><i><font color="Green" title="E7">A2</font></i>: 
 <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">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" title="RADIX_2:th.6">RADIX_2:6</a></i>;<br/>
<a NAME="E3:5"/><i><font color="Green" title="E8">A3</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="radix_5.html#E1:5"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> 0,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c1">i</font> = 
 <a href="radix_1.html#K9" title="RADIX_1:func.9">DigitDC</a> 0,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">k</font>
<b>by </b><i><a class="txt" href="radix_5.html#E1:5"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="radix_1.html#D9" title="RADIX_1:def.9">RADIX_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default">0 <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> <font color="Maroon" title="c1">i</font></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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="radix_1.html#D8" title="RADIX_1:def.8">RADIX_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default">0 <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="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">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="c2">k</font></span>)</span>
<b>by </b><i><a class="txt" href="radix_5.html#E2:5"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="radix_5.html#E3:5"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="radix_2.html#T4" title="RADIX_2:th.4">RADIX_2:4</a></i>
<br/>.= 
0 <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="nat_2.html#T4" title="NAT_2:th.4">NAT_2:4</a></i>
;<br/>
<b>hence </b><a NAME="E5:5"/>
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> 0,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_d.html#T26" title="NAT_D:th.26">NAT_D:26</a></i>;<br/>


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