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<span class="kw">let </span><font color="Maroon" title="c1">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">tx</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of 1,<font color="Maroon" title="c1">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a>  holds   <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Olive" title="b1">tx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<font color="Olive" title="b1">tx</font>,1)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">tx</font> be   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of 1,<font color="Maroon" title="c1">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Maroon" title="c2">tx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<font color="Maroon" title="c2">tx</font>,1)</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c3">w</font> =  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<font color="Maroon" title="c2">tx</font>,1) as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></span>;<br/>
<a NAME="E2:4"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E6">A2</font></span>: <span class="p1">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <font color="Maroon" title="c2">tx</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
 <a href="radix_1.html#K6" title="RADIX_1:func.6">SubDigit</a> (<font color="Maroon" title="c2">tx</font>,1,<font color="Maroon" title="c1">k</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_1.html#D6" title="RADIX_1:def.6">RADIX_1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K5" title="RADIX_1:func.5">DigB</a> (<font color="Maroon" title="c2">tx</font>,1)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_1.html#D5" title="RADIX_1:def.5">RADIX_1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<font color="Maroon" title="c2">tx</font>,1)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_1.html#D4" title="RADIX_1:def.4">RADIX_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<font color="Maroon" title="c2">tx</font>,1)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T232" title="XREAL_1:th.232">XREAL_1:232</a></span>
<br/>.= 
1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<font color="Maroon" title="c2">tx</font>,1)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T4" title="NEWTON:th.4">NEWTON:4</a></span>
;<br/>
<a NAME="E4:4"/><span class="lab"><font color="Green" title="E7">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <font color="Maroon" title="c2">tx</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<a NAME="E5:4"/><span class="kw">then </span>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <font color="Maroon" title="c2">tx</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:4"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E6:4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A4</font></span>: 
<span class="p1">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <font color="Maroon" title="c2">tx</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<font color="Maroon" title="c2">tx</font>,1)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:4"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
 <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Maroon" title="c2">tx</font> = 
 <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <span class="p1">(<span class="default"><a href="radix_1.html#K7" title="RADIX_1:func.7">DigitSD</a> <font color="Maroon" title="c2">tx</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_1.html#D7" title="RADIX_1:def.7">RADIX_1:def 7</a></span>
<br/>.= 
 <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:4"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="txt" href="#E6:4"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>
<br/>.= 
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<font color="Maroon" title="c2">tx</font>,1)
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T73" title="RVSUM_1:th.73">RVSUM_1:73</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:4"/>
 <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Maroon" title="c2">tx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<font color="Maroon" title="c2">tx</font>,1)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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