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<span class="kw">let </span><font color="Maroon" title="c1">k</font>, <font color="Maroon" title="c2">m</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"> ( 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c2">m</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> 1,<font color="Maroon" title="c1">k</font> implies  <a href="radix_3.html#K7" title="RADIX_3:func.7">DigA_SDSub</a> (<span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c2">m</font>,1,<font color="Maroon" title="c1">k</font>)</span>)</span></span>)</span>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">k</font>) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:7"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">and </span><br/><a NAME="E2:7"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> 1,<font color="Maroon" title="c1">k</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_3.html#K7" title="RADIX_3:func.7">DigA_SDSub</a> (<span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c2">m</font>,1,<font color="Maroon" title="c1">k</font>)</span>)</span></span>)</span>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">k</font>)</span><br/>

<span class="kw">thus </span> <a href="radix_3.html#K7" title="RADIX_3:func.7">DigA_SDSub</a> (<span class="p1">(<span class="default"><a href="radix_3.html#K10" title="RADIX_3:func.10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c2">m</font>,1,<font color="Maroon" title="c1">k</font>)</span>)</span></span>)</span>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) = 
 <a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_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> (<font color="Maroon" title="c2">m</font>,1,<font color="Maroon" title="c1">k</font>)</span>)</span>,1)</span>)</span>,<font color="Maroon" title="c1">k</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="radix_4.html#T3" target="_self" title="RADIX_4:th.3">Th3</a></span>
<br/>.= 
 <a href="radix_3.html#K5" title="RADIX_3:func.5">SDSub_Add_Carry</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">k</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="radix_1.html#T21" title="RADIX_1:th.21">RADIX_1:21</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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