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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c1">n</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">y</font> being  <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c1">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a> </span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> holds <br/> <a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <font color="Olive" title="b2">y</font></span><br/><b>let </b><font color="Maroon" title="c3">x</font> be   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c1">n</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">y</font> being  <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c1">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a> </span>)</span>  st <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> holds <br/> <a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <font color="Olive" title="b1">y</font></span><br/><b>let </b><font color="Maroon" title="c4">y</font> be   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c1">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a> </span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> implies  <a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <font color="Maroon" title="c4">y</font> )</span><br/>





<b>assume </b><a NAME="E1:17"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <font color="Maroon" title="c4">y</font></span><br/>

<div><i><font color="Green" title="E14">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c3">x</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K5" title="RADIX_1:func.5">DigB</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">i</font> )</span><br/><b>assume </b><a NAME="E1:17_1"/>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K5" title="RADIX_1:func.5">DigB</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">i</font></span><br/><b>then </b><font color="Maroon" title="c3">x</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">i</font> = 
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">i</font>
<b>by </b><i><a class="txt" href="#E1:17"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="radix_1.html#D3" title="RADIX_1:def.3">RADIX_1:def 3</a></i>
<br/>.= 
 <a href="radix_1.html#K5" title="RADIX_1:func.5">DigB</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">i</font>
<b>by </b><i><a class="ref" href="radix_1.html#D4" title="RADIX_1:def.4">RADIX_1:def 4</a></i>
;<br/><b>hence </b><a NAME="E3:17_1"/>
<font color="Maroon" title="c3">x</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K5" title="RADIX_1:func.5">DigB</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E3:17"/><i><font color="Green" title="E15">A3</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E4:17"/><b>then </b><i><font color="Green" title="E16">X</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">k</font></span>)</span> implies <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">j</font> )</span><br/><b>assume </b><a NAME="E1:17_2"/><i><font color="Green" title="E17">A4</font></i>: 
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">k</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">j</font></span><br/><a NAME="E2:17_2"/><b>then </b><i><font color="Green" title="E18">A5</font></i>: 
( <font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">k</font></span>)</span> &amp; <font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <font color="Maroon" title="c4">y</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E3:17"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="#E4:17"><i><font color="Green" title="E16">X</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">j</font> = 
<span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font>
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
 <a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c2">k</font>
<b>by </b><i><a class="txt" href="#E1:17_2"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="radix_2.html#D2" target="_self" title="RADIX_2:def.2">Def2</a>, <a class="txt" href="#E4:17"><i><font color="Green" title="E16">X</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K5" title="RADIX_1:func.5">DigB</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">j</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:17"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="#E1:17_2"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="#E4:17"><i><font color="Green" title="E16">X</font></i></a></i>
<br/>.= 
 <a href="radix_1.html#K6" title="RADIX_1:func.6">SubDigit</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c2">k</font>
<b>by </b><i><a class="ref" href="radix_1.html#D5" title="RADIX_1:def.5">RADIX_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font>
<b>by </b><i><a class="txt" href="#E1:17_2"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="#E4:17"><i><font color="Green" title="E16">X</font></i></a>, <a class="ref" href="radix_1.html#D6" title="RADIX_1:def.6">RADIX_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">j</font>
<b>by </b><i><a class="txt" href="#E2:17_2"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/><b>hence </b><a NAME="E4:17_2"/>
<span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:17"/>
 <a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="#E3:17"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
