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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 holds <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">a<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="radix_1.html#K3">-SD</a> </span>)</span> st ( for <font color="Olive">b<sub>6</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>(  not <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font> or (  <a href="radix_1.html#K4">DigA</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>6</sub></font> &amp;  <a href="radix_1.html#K4">DigA</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>6</sub></font> ) or (  <a href="radix_1.html#K4">DigA</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>6</sub></font> &amp;  <a href="radix_1.html#K4">DigA</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>6</sub></font> ) ) ) holds <br/><span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Olive">b<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Olive">b<sub>3</sub></font></span>)</span>;<br/>
<a NAME="E1:16"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:16_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="binarith.html#NM1">Tuple</a> of 1,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="radix_1.html#K3">-SD</a> </span>)</span>;<br/>

<b>assume </b><a NAME="E2:16_1"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1 or (  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp;  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font> ) or (  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp;  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>

<a NAME="E3:16_1"/><i><font color="Green">E17</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<i><font color="Green">E18</font></i>:  <a href="radix_1.html#K8">SDDec</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="radix_1.html#K14">'+'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="radix_1.html#K14">'+'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,1
<b>by </b><i><a class="ref" href="radix_5.html#T4" target="_self">Th4</a></i>
<br/>.= 
 <a href="radix_1.html#K13">Add</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,1,<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="radix_5.html#E3:16_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="radix_1.html#D14">RADIX_1:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K12">SD_Add_Data</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K11">SD_Add_Carry</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default">1 <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p4">(<span class="default">1 <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="radix_5.html#E1:16_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="radix_5.html#E3:16_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="radix_1.html#D13">RADIX_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K12">SD_Add_Data</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K11">SD_Add_Carry</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>4</sub></font>,0</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p4">(<span class="default">1 <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K12">SD_Add_Data</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K11">SD_Add_Carry</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>4</sub></font>,0</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>5</sub></font>,0</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K12">SD_Add_Data</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K11">SD_Add_Carry</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>4</sub></font>,0</span>)</span> <a href="xcmplx_0.html#K2">+</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="radix_1.html#D3">RADIX_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K12">SD_Add_Data</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K11">SD_Add_Carry</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="radix_1.html#D3">RADIX_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K12">SD_Add_Data</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0
<b>by </b><i><a class="ref" href="radix_1.html#D10">RADIX_1:def 10</a></i>
;<br/>
<i><font color="Green">E19</font></i>:  <a href="radix_1.html#K8">SDDec</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="radix_1.html#K14">'+'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
 <a href="radix_1.html#K4">DigA</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="radix_1.html#K14">'+'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,1
<b>by </b><i><a class="ref" href="radix_5.html#T4" target="_self">Th4</a></i>
<br/>.= 
 <a href="radix_1.html#K13">Add</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,1,<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="radix_5.html#E3:16_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="radix_1.html#D14">RADIX_1:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K12">SD_Add_Data</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>2</sub></font>,1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K11">SD_Add_Carry</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p4">(<span class="default">1 <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default">1 <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="radix_5.html#E1:16_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="radix_5.html#E3:16_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="radix_1.html#D13">RADIX_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K12">SD_Add_Data</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>2</sub></font>,1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K11">SD_Add_Carry</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>2</sub></font>,0</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default">1 <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K12">SD_Add_Data</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>2</sub></font>,1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K11">SD_Add_Carry</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>2</sub></font>,0</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,0</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K12">SD_Add_Data</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>2</sub></font>,1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K11">SD_Add_Carry</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>2</sub></font>,0</span>)</span> <a href="xcmplx_0.html#K2">+</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="radix_1.html#D3">RADIX_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K12">SD_Add_Data</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>2</sub></font>,1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K11">SD_Add_Carry</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="radix_1.html#D3">RADIX_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K12">SD_Add_Data</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>2</sub></font>,1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0
<b>by </b><i><a class="ref" href="radix_1.html#D10">RADIX_1:def 10</a></i>
;<br/>
<a NAME="E6:16_1"/><i><font color="Green">E20</font></i>: 
( (  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>2</sub></font>,1 <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>4</sub></font>,1 &amp;  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,1 <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>5</sub></font>,1 ) or (  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,1 <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>4</sub></font>,1 &amp;  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>2</sub></font>,1 <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>5</sub></font>,1 ) )
 
<b>by </b><i><a class="txt" href="radix_5.html#E2:16_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="radix_5.html#E3:16_1"><i><font color="Green">E17</font></i></a></i>;<br/>
<a NAME="E7:16_1"/><i><font color="Green">E21</font></i>: 
(  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>2</sub></font>,1 <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>4</sub></font>,1 &amp;  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,1 <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>5</sub></font>,1 implies <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_3"/>
(  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>2</sub></font>,1 <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>4</sub></font>,1 &amp;  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,1 <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>5</sub></font>,1 )
 ;<br/>

<a NAME="E2:16_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="radix_1.html#K14">'+'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K11">SD_Add_Carry</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>2</sub></font>,1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wsierp_1.html#K2">|^</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="radix_5.html#E1:16_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="radix_5.html#E4:16_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="radix_5.html#E5:16_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="radix_2.html#T11">RADIX_2:11</a></i>;<br/>
<b>hence </b><a NAME="E3:16_1_3"/>
<span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="radix_5.html#E1:16_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="radix_2.html#T11">RADIX_2:11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:16_1"/>
(  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,1 <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>4</sub></font>,1 &amp;  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>2</sub></font>,1 <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>5</sub></font>,1 implies <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_4"/>
(  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,1 <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>4</sub></font>,1 &amp;  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>2</sub></font>,1 <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>5</sub></font>,1 )
 ;<br/>

<a NAME="E2:16_1_4"/>
<span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="radix_1.html#K14">'+'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K11">SD_Add_Carry</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>2</sub></font>,1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wsierp_1.html#K2">|^</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="radix_5.html#E1:16_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="radix_5.html#E4:16_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="radix_5.html#E5:16_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="radix_5.html#E6:16_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="radix_2.html#T11">RADIX_2:11</a></i>;<br/>
<b>hence </b><a NAME="E3:16_1_4"/>
<span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="radix_5.html#E1:16_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="radix_2.html#T11">RADIX_2:11</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:16_1"/>
<span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="radix_5.html#E2:16_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="radix_5.html#E3:16_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="radix_5.html#E7:16_1"><i><font color="Green">E21</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:16"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:16_2"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>] )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E2:16_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="binarith.html#NM1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="radix_1.html#K3">-SD</a> </span>)</span>;<br/>

<b>assume </b><a NAME="E3:16_2"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> or (  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp;  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font> ) or (  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp;  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="wsierp_1.html#K6">INT</a>  =  <a href="radix_1.html#K5">DigB</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="wsierp_1.html#K6">INT</a> <b> such that </b><br/><a NAME="E5:16_2"/><i><font color="Green">E19</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E6:16_2"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<a NAME="E7:16_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="radix_1.html#K3">-SD</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:16_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:16_2_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E4:16_2_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="radix_5.html#E3:16_2_1"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E6:16_2_1"/><i><font color="Green">E23</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="radix_5.html#E5:16_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then </b><i><font color="Green">E24</font></i>: <font color="Maroon">c<sub>8</sub></font> = 
 <a href="radix_1.html#K5">DigB</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="radix_5.html#E6:16_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="radix_5.html#E3:16_2_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="radix_5.html#E4:16_2_1"><i><font color="Green">E22</font></i></a></i>
<br/>.= 
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="radix_1.html#D4">RADIX_1:def 4</a></i>
;<br/>
<a NAME="E8:16_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="radix_5.html#E3:16_2_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="radix_5.html#E6:16_2_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_2.html#T9">FINSEQ_2:9</a></i>;<br/>
<a NAME="E9:16_2_1"/><b>then </b>
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="radix_1.html#K3">-SD</a> 
 
<b>by </b><i><a class="ref" href="radix_1.html#T19">RADIX_1:19</a></i>;<br/>
<b>hence </b><a NAME="E10:16_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="radix_1.html#K3">-SD</a> 
 <b>by </b><i><a class="txt" href="radix_5.html#E7:16_2_1"><i><font color="Green">E24</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>2</sub></font> <a href="radix_1.html#K3">-SD</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E9:16_2"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:16_2_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:16_2_2"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T9">FINSEQ_2:9</a></i>;<br/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> = 
 <a href="radix_1.html#K5">DigB</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="radix_5.html#E6:16_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="radix_5.html#E1:16_2_2"><i><font color="Green">E22</font></i></a></i>
<br/>.= 
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="radix_1.html#D4">RADIX_1:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E4:16_2_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="radix_5.html#E2:16_2_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="radix_1.html#D3">RADIX_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="wsierp_1.html#K6">INT</a>  =  <a href="radix_1.html#K5">DigB</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="wsierp_1.html#K6">INT</a> <b> such that </b><br/><a NAME="E11:16_2"/><i><font color="Green">E22</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E12:16_2"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<a NAME="E13:16_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="radix_1.html#K3">-SD</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:16_2_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:16_2_3"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E4:16_2_3"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="radix_5.html#E3:16_2_3"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E6:16_2_3"/><i><font color="Green">E26</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="radix_5.html#E11:16_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then </b><i><font color="Green">E27</font></i>: <font color="Maroon">c<sub>10</sub></font> = 
 <a href="radix_1.html#K5">DigB</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="radix_5.html#E12:16_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="radix_5.html#E3:16_2_3"><i><font color="Green">E24</font></i></a>, <a class="txt" href="radix_5.html#E4:16_2_3"><i><font color="Green">E25</font></i></a></i>
<br/>.= 
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="radix_1.html#D4">RADIX_1:def 4</a></i>
;<br/>
<a NAME="E8:16_2_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="radix_5.html#E3:16_2_3"><i><font color="Green">E24</font></i></a>, <a class="txt" href="radix_5.html#E6:16_2_3"><i><font color="Green">E26</font></i></a>, <a class="ref" href="finseq_2.html#T9">FINSEQ_2:9</a></i>;<br/>
<a NAME="E9:16_2_3"/><b>then </b>
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>12</sub></font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="radix_1.html#K3">-SD</a> 
 
<b>by </b><i><a class="ref" href="radix_1.html#T19">RADIX_1:19</a></i>;<br/>
<b>hence </b><a NAME="E10:16_2_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="radix_1.html#K3">-SD</a> 
 <b>by </b><i><a class="txt" href="radix_5.html#E7:16_2_3"><i><font color="Green">E27</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>2</sub></font> <a href="radix_1.html#K3">-SD</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E15:16_2"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:16_2_4"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:16_2_4"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T9">FINSEQ_2:9</a></i>;<br/>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> = 
 <a href="radix_1.html#K5">DigB</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="radix_5.html#E12:16_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="radix_5.html#E1:16_2_4"><i><font color="Green">E25</font></i></a></i>
<br/>.= 
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="radix_1.html#D4">RADIX_1:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E4:16_2_4"/>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="radix_5.html#E2:16_2_4"><i><font color="Green">E26</font></i></a>, <a class="ref" href="radix_1.html#D3">RADIX_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="wsierp_1.html#K6">INT</a>  =  <a href="radix_1.html#K5">DigB</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="wsierp_1.html#K6">INT</a> <b> such that </b><br/><a NAME="E17:16_2"/><i><font color="Green">E25</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E18:16_2"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<a NAME="E19:16_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="radix_1.html#K3">-SD</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:16_2_5"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:16_2_5"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E4:16_2_5"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="radix_5.html#E3:16_2_5"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E6:16_2_5"/><i><font color="Green">E29</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="radix_5.html#E17:16_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then </b><i><font color="Green">E30</font></i>: <font color="Maroon">c<sub>12</sub></font> = 
 <a href="radix_1.html#K5">DigB</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="radix_5.html#E18:16_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="radix_5.html#E3:16_2_5"><i><font color="Green">E27</font></i></a>, <a class="txt" href="radix_5.html#E4:16_2_5"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="radix_1.html#D4">RADIX_1:def 4</a></i>
;<br/>
<a NAME="E8:16_2_5"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="radix_5.html#E3:16_2_5"><i><font color="Green">E27</font></i></a>, <a class="txt" href="radix_5.html#E6:16_2_5"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_2.html#T9">FINSEQ_2:9</a></i>;<br/>
<a NAME="E9:16_2_5"/><b>then </b>
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>14</sub></font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="radix_1.html#K3">-SD</a> 
 
<b>by </b><i><a class="ref" href="radix_1.html#T19">RADIX_1:19</a></i>;<br/>
<b>hence </b><a NAME="E10:16_2_5"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="radix_1.html#K3">-SD</a> 
 <b>by </b><i><a class="txt" href="radix_5.html#E7:16_2_5"><i><font color="Green">E30</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>2</sub></font> <a href="radix_1.html#K3">-SD</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E21:16_2"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:16_2_6"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:16_2_6"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T9">FINSEQ_2:9</a></i>;<br/>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> = 
 <a href="radix_1.html#K5">DigB</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="radix_5.html#E18:16_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="radix_5.html#E1:16_2_6"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="radix_1.html#D4">RADIX_1:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E4:16_2_6"/>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="radix_5.html#E2:16_2_6"><i><font color="Green">E29</font></i></a>, <a class="ref" href="radix_1.html#D3">RADIX_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="wsierp_1.html#K6">INT</a>  =  <a href="radix_1.html#K5">DigB</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="wsierp_1.html#K6">INT</a> <b> such that </b><br/><a NAME="E23:16_2"/><i><font color="Green">E28</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E24:16_2"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<a NAME="E25:16_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="radix_1.html#K3">-SD</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:16_2_7"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:16_2_7"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E4:16_2_7"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="radix_5.html#E3:16_2_7"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E6:16_2_7"/><i><font color="Green">E32</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="radix_5.html#E23:16_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then </b><i><font color="Green">E33</font></i>: <font color="Maroon">c<sub>14</sub></font> = 
 <a href="radix_1.html#K5">DigB</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="txt" href="radix_5.html#E24:16_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="radix_5.html#E3:16_2_7"><i><font color="Green">E30</font></i></a>, <a class="txt" href="radix_5.html#E4:16_2_7"><i><font color="Green">E31</font></i></a></i>
<br/>.= 
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="ref" href="radix_1.html#D4">RADIX_1:def 4</a></i>
;<br/>
<a NAME="E8:16_2_7"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="radix_5.html#E3:16_2_7"><i><font color="Green">E30</font></i></a>, <a class="txt" href="radix_5.html#E6:16_2_7"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_2.html#T9">FINSEQ_2:9</a></i>;<br/>
<a NAME="E9:16_2_7"/><b>then </b>
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>16</sub></font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="radix_1.html#K3">-SD</a> 
 
<b>by </b><i><a class="ref" href="radix_1.html#T19">RADIX_1:19</a></i>;<br/>
<b>hence </b><a NAME="E10:16_2_7"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="radix_1.html#K3">-SD</a> 
 <b>by </b><i><a class="txt" href="radix_5.html#E7:16_2_7"><i><font color="Green">E33</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>2</sub></font> <a href="radix_1.html#K3">-SD</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E27:16_2"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:16_2_8"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:16_2_8"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T9">FINSEQ_2:9</a></i>;<br/>
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> = 
 <a href="radix_1.html#K5">DigB</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="radix_5.html#E24:16_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="radix_5.html#E1:16_2_8"><i><font color="Green">E31</font></i></a></i>
<br/>.= 
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="ref" href="radix_1.html#D4">RADIX_1:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E4:16_2_8"/>
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="radix_5.html#E2:16_2_8"><i><font color="Green">E32</font></i></a>, <a class="ref" href="radix_1.html#D3">RADIX_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="radix_1.html#K3">-SD</a> </span>)</span> <b>by </b><i><a class="txt" href="radix_5.html#E5:16_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="radix_1.html#K3">-SD</a> </span>)</span> <b>by </b><i><a class="txt" href="radix_5.html#E11:16_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="radix_1.html#K3">-SD</a> </span>)</span> <b>by </b><i><a class="txt" href="radix_5.html#E17:16_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="radix_1.html#K3">-SD</a> </span>)</span> <b>by </b><i><a class="txt" href="radix_5.html#E23:16_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E32:16_2"/><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="radix_5.html#E9:16_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="radix_2.html#T10">RADIX_2:10</a></i>;<br/>
<a NAME="E33:16_2"/><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="radix_5.html#E15:16_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="radix_2.html#T10">RADIX_2:10</a></i>;<br/>
<a NAME="E34:16_2"/><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="radix_5.html#E21:16_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="radix_2.html#T10">RADIX_2:10</a></i>;<br/>
<a NAME="E35:16_2"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="radix_5.html#E27:16_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="radix_2.html#T10">RADIX_2:10</a></i>;<br/>
<a NAME="E36:16_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> or (  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp;  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Olive">b<sub>1</sub></font> ) or (  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp;  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Olive">b<sub>1</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:16_2_9"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:16_2_9"/><b>then </b><i><font color="Green">E36</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:16_2_9"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E4:16_2_9"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="radix_5.html#E2:16_2_9"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>then </b><i><font color="Green">E38</font></i>:  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>19</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="ref" href="radix_1.html#D3">RADIX_1:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="radix_5.html#E9:16_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="radix_5.html#E1:16_2_9"><i><font color="Green">E35</font></i></a></i>
<br/>.= 
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="radix_5.html#E1:16_2_9"><i><font color="Green">E35</font></i></a>, <a class="ref" href="radix_1.html#D3">RADIX_1:def 3</a></i>
;<br/>
<i><font color="Green">E39</font></i>:  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>19</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="radix_5.html#E4:16_2_9"><i><font color="Green">E37</font></i></a>, <a class="ref" href="radix_1.html#D3">RADIX_1:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="radix_5.html#E15:16_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="radix_5.html#E1:16_2_9"><i><font color="Green">E35</font></i></a></i>
<br/>.= 
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="radix_5.html#E1:16_2_9"><i><font color="Green">E35</font></i></a>, <a class="ref" href="radix_1.html#D3">RADIX_1:def 3</a></i>
;<br/>
<i><font color="Green">E40</font></i>:  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>19</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="radix_5.html#E4:16_2_9"><i><font color="Green">E37</font></i></a>, <a class="ref" href="radix_1.html#D3">RADIX_1:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="radix_5.html#E21:16_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="radix_5.html#E1:16_2_9"><i><font color="Green">E35</font></i></a></i>
<br/>.= 
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="radix_5.html#E1:16_2_9"><i><font color="Green">E35</font></i></a>, <a class="ref" href="radix_1.html#D3">RADIX_1:def 3</a></i>
;<br/>
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>19</sub></font> = 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="radix_5.html#E4:16_2_9"><i><font color="Green">E37</font></i></a>, <a class="ref" href="radix_1.html#D3">RADIX_1:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="radix_5.html#E27:16_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="radix_5.html#E1:16_2_9"><i><font color="Green">E35</font></i></a></i>
<br/>.= 
 <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="radix_5.html#E1:16_2_9"><i><font color="Green">E35</font></i></a>, <a class="ref" href="radix_1.html#D3">RADIX_1:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E9:16_2_9"/>
( (  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font> &amp;  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font> ) or (  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font> &amp;  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font> ) )
 <b>by </b><i><a class="txt" href="radix_5.html#E3:16_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="radix_5.html#E4:16_2_9"><i><font color="Green">E37</font></i></a>, <a class="txt" href="radix_5.html#E5:16_2_9"><i><font color="Green">E38</font></i></a>, <a class="txt" href="radix_5.html#E6:16_2_9"><i><font color="Green">E39</font></i></a>, <a class="txt" href="radix_5.html#E7:16_2_9"><i><font color="Green">E40</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E37:16_2"/><b>then </b><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="radix_5.html#E1:16_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="radix_5.html#E2:16_2"><i><font color="Green">E17</font></i></a></i>;<br/>
<i><font color="Green">E36</font></i>: <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="radix_5.html#E34:16_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="radix_5.html#E35:16_2"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>

;<br/>
<i><font color="Green">E37</font></i>: <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="radix_5.html#E32:16_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="radix_5.html#E33:16_2"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="radix_5.html#E37:16_2"><i><font color="Green">E35</font></i></a></i>
;<br/>
<a NAME="E40:16_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E41:16_2"/><b>then </b>
( (  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or (  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) )
 
<b>by </b><i><a class="txt" href="radix_5.html#E3:16_2"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>hence </b><a NAME="E42:16_2"/>
<span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="radix_5.html#E38:16_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="radix_5.html#E39:16_2"><i><font color="Green">E37</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:16"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S9">NAT_1:sch 9</a>(<a class="txt" href="radix_5.html#E1:16"><i><font color="Green">E14</font></i></a>, <a class="txt" href="radix_5.html#E2:16"><i><font color="Green">E15</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:16"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 holds <br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></font> being  <a href="binarith.html#NM1">Tuple</a> of <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="radix_1.html#K3">-SD</a> </span>)</span> st ( for <font color="Olive">b<sub>7</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>(  not <font color="Olive">b<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Olive">b<sub>1</sub></font> or (  <a href="radix_1.html#K4">DigA</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>7</sub></font> &amp;  <a href="radix_1.html#K4">DigA</a> <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>7</sub></font> ) or (  <a href="radix_1.html#K4">DigA</a> <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>7</sub></font> &amp;  <a href="radix_1.html#K4">DigA</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="radix_1.html#K4">DigA</a> <font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>7</sub></font> ) ) ) holds <br/><span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Olive">b<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Olive">b<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_1.html#K8">SDDec</a> <font color="Olive">b<sub>4</sub></font></span>)</span>
 ;<br/>


</div>
