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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">k</font>, <font color="Maroon" title="c2">m</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &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> &amp; <font color="Maroon" title="c3">n</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_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p1">(<span class="default"><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> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">n</font>,1,<font color="Maroon" title="c1">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K12" title="RADIX_1:func.12">SD_Add_Data</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c1">k</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:44"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <span class="kw">and </span><br/><a NAME="E2:44"/><span class="lab"><font color="Green" title="E32">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>
 <span class="kw">and </span><br/><a NAME="E3:44"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
<font color="Maroon" title="c3">n</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="Red">::  thesis: </font></span></a><span class="hide">  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p1">(<span class="default"><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> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">n</font>,1,<font color="Maroon" title="c1">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K12" title="RADIX_1:func.12">SD_Add_Data</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c1">k</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">N</font> =  <a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">n</font>,1,<font color="Maroon" title="c1">k</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">M</font> =  <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>;<br/>
<a NAME="E4:44"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E35">A5</font></span>:  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p1">(<span class="default"><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> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">n</font>,1,<font color="Maroon" title="c1">k</font></span>)</span></span>)</span>,1 = 
 <a href="radix_1.html#K13" title="RADIX_1:func.13">Add</a> <span class="p1">(<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 class="p1">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">n</font>,1,<font color="Maroon" title="c1">k</font></span>)</span>,1,<font color="Maroon" title="c1">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_1.html#D14" target="_self" title="RADIX_1:def.14">Def14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K12" title="RADIX_1:func.12">SD_Add_Data</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p4">(<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> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">n</font>,1,<font color="Maroon" title="c1">k</font></span>)</span>,1</span>)</span></span>)</span>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K11" title="RADIX_1:func.11">SD_Add_Carry</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p4">(<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 class="p4">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">n</font>,1,<font color="Maroon" title="c1">k</font></span>)</span>,<span class="p4">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E4:44"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="ref" href="radix_1.html#D13" target="_self" title="RADIX_1:def.13">Def13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K12" title="RADIX_1:func.12">SD_Add_Data</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p4">(<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> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">n</font>,1,<font color="Maroon" title="c1">k</font></span>)</span>,1</span>)</span></span>)</span>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K11" title="RADIX_1:func.11">SD_Add_Carry</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p4">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">n</font>,1,<font color="Maroon" title="c1">k</font></span>)</span>,<span class="p4">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T234" title="XREAL_1:th.234">XREAL_1:234</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K12" title="RADIX_1:func.12">SD_Add_Data</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p4">(<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> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">n</font>,1,<font color="Maroon" title="c1">k</font></span>)</span>,1</span>)</span></span>)</span>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K11" title="RADIX_1:func.11">SD_Add_Carry</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p4">(<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">n</font>,1,<font color="Maroon" title="c1">k</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T234" title="XREAL_1:th.234">XREAL_1:234</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K12" title="RADIX_1:func.12">SD_Add_Data</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p4">(<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> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">n</font>,1,<font color="Maroon" title="c1">k</font></span>)</span>,1</span>)</span></span>)</span>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K11" title="RADIX_1:func.11">SD_Add_Carry</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">n</font>,1,<font color="Maroon" title="c1">k</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_1.html#D3" target="_self" title="RADIX_1:def.3">Def3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K12" title="RADIX_1:func.12">SD_Add_Data</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p4">(<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> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p4">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">n</font>,1,<font color="Maroon" title="c1">k</font></span>)</span>,1</span>)</span></span>)</span>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_1.html#D3" target="_self" title="RADIX_1:def.3">Def3</a>, <a class="ref" href="radix_1.html#T21" target="_self" title="RADIX_1:th.21">Th21</a></span>
<br/>.= 
 <a href="radix_1.html#K12" title="RADIX_1:func.12">SD_Add_Data</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">n</font>,1,<font color="Maroon" title="c1">k</font></span>)</span>,1</span>)</span></span>)</span>,<font color="Maroon" title="c1">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="radix_1.html#T24" target="_self" title="RADIX_1:th.24">Th24</a></span>
<br/>.= 
 <a href="radix_1.html#K12" title="RADIX_1:func.12">SD_Add_Data</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:44"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="radix_1.html#T24" target="_self" title="RADIX_1:th.24">Th24</a></span>
;<br/>
<span class="lab"><font color="Green" title="E36">A6</font></span>: <span class="p1">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <span class="p2">(<span class="default"><span class="p3">(<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> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">n</font>,1,<font color="Maroon" title="c1">k</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
 <a href="radix_1.html#K6" title="RADIX_1:func.6">SubDigit</a> <span class="p1">(<span class="default"><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> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">n</font>,1,<font color="Maroon" title="c1">k</font></span>)</span></span>)</span>,1,<font color="Maroon" title="c1">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:44"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="ref" href="radix_1.html#D6" target="_self" title="RADIX_1:def.6">Def6</a></span>
<br/>.= 
<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="c1">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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"><span class="p3">(<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> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">n</font>,1,<font color="Maroon" title="c1">k</font></span>)</span></span>)</span>,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T234" title="XREAL_1:th.234">XREAL_1:234</a></span>
<br/>.= 
1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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"><span class="p3">(<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> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">n</font>,1,<font color="Maroon" title="c1">k</font></span>)</span></span>)</span>,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></span>
<br/>.= 
 <a href="radix_1.html#K12" title="RADIX_1:func.12">SD_Add_Data</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:44"><span class="lab"><font color="Green" title="E35">A5</font></span></a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">w</font> =  <a href="radix_1.html#K12" title="RADIX_1:func.12">SD_Add_Data</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c1">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></span>;<br/>
<a NAME="E8:44"/><span class="lab"><font color="Green" title="E37">A7</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <span class="p2">(<span class="default"><span class="p3">(<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> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">n</font>,1,<font color="Maroon" title="c1">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<a NAME="E9:44"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E10:44"/><span class="kw">then </span>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <span class="p2">(<span class="default"><span class="p3">(<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> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">n</font>,1,<font color="Maroon" title="c1">k</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:44"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E11:44"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <span class="p2">(<span class="default"><span class="p3">(<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> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">n</font>,1,<font color="Maroon" title="c1">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K12" title="RADIX_1:func.12">SD_Add_Data</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:44"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/>
<span class="kw">then </span> <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p1">(<span class="default"><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> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">n</font>,1,<font color="Maroon" title="c1">k</font></span>)</span></span>)</span> = 
 <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">w</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:44"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>
<br/>.= 
 <a href="radix_1.html#K12" title="RADIX_1:func.12">SD_Add_Data</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T103" title="RVSUM_1:th.103">RVSUM_1:103</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E13:44"/>
 <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p1">(<span class="default"><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> <a href="radix_1.html#K14" title="RADIX_1:func.14">'+'</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> <font color="Maroon" title="c3">n</font>,1,<font color="Maroon" title="c1">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K12" title="RADIX_1:func.12">SD_Add_Data</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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