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<span class="kw">let </span><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">k</font>, <font color="Maroon" title="c3">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> implies  <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> (<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:5"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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> (<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E2:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A2</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<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> (<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<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> (<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">k</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="radix_1.html#D9" title="RADIX_1:def.9">RADIX_1:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <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#K2" title="NEWTON:func.2">|^</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#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_1.html#D8" title="RADIX_1:def.8">RADIX_1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <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#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="radix_2.html#T4" title="RADIX_2:th.4">RADIX_2:4</a>, <a class="ref" href="radix_2.html#T6" title="RADIX_2:th.6">RADIX_2:6</a></span>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T2" title="NAT_2:th.2">NAT_2:2</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:5"/>
 <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> (<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T26" title="NAT_D:th.26">NAT_D:26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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