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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">k</font>, <font color="Maroon" title="c3">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Maroon" title="c3">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="c1">n</font> implies  <a href="radix_1.html#K11" title="RADIX_1:func.11">SD_Add_Carry</a> <span class="p1">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> 1,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<b>assume </b><a NAME="E1:13"/><i><font color="Green" title="E12">A1</font></i>: 
( <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Maroon" title="c3">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="c1">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="radix_1.html#K11" title="RADIX_1:func.11">SD_Add_Carry</a> <span class="p1">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> 1,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c3">i</font></span>)</span> <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="E2:13"/><b>then </b><i><font color="Green" title="E13">A2</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_1_1"/>
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><b>by </b><i><a class="txt" href="#E2:13"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_1"><b>suppose </b></a><a NAME="E1:13_1_1_1"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="radix_1.html#K11" title="RADIX_1:func.11">SD_Add_Carry</a> <span class="p1">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> 1,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><a NAME="E2:13_1_1_1"/><b>then </b>
 <a href="radix_1.html#K11" title="RADIX_1:func.11">SD_Add_Carry</a> <span class="p1">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> 1,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K11" title="RADIX_1:func.11">SD_Add_Carry</a> 1
 <b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="radix_5.html#T7" target="_self" title="RADIX_5:th.7">Th7</a></i>;<br/><b>hence </b><a NAME="E3:13_1_1_1"/>
 <a href="radix_1.html#K11" title="RADIX_1:func.11">SD_Add_Carry</a> <span class="p1">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> 1,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="radix_1.html#D10" title="RADIX_1:def.10">RADIX_1:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_2"><b>suppose </b></a><a NAME="E1:13_1_1_2"/>
<font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="radix_1.html#K11" title="RADIX_1:func.11">SD_Add_Carry</a> <span class="p1">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> 1,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><a NAME="E2:13_1_1_2"/><b>then </b>
 <a href="radix_1.html#K11" title="RADIX_1:func.11">SD_Add_Carry</a> <span class="p1">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> 1,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K11" title="RADIX_1:func.11">SD_Add_Carry</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E1:13"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="radix_5.html#T8" target="_self" title="RADIX_5:th.8">Th8</a></i>;<br/><b>hence </b><a NAME="E3:13_1_1_2"/>
 <a href="radix_1.html#K11" title="RADIX_1:func.11">SD_Add_Carry</a> <span class="p1">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> 1,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="radix_1.html#D10" title="RADIX_1:def.10">RADIX_1:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:13"/>
 <a href="radix_1.html#K11" title="RADIX_1:func.11">SD_Add_Carry</a> <span class="p1">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> 1,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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