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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">q</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><br/>  for <font color="Olive" title="b2">ic</font>, <font color="Olive" title="b3">f</font>, <font color="Olive" title="b4">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">ic</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> $1,<font color="Olive" title="b4">k</font> &amp; <font color="Olive" title="b3">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> for <font color="Olive" title="b5">c</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of $1,<font color="Olive" title="b4">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a>   st <font color="Olive" title="b5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Olive" title="b2">ic</font>,$1,<font color="Olive" title="b4">k</font>) holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Olive" title="b1">q</font>,<font color="Olive" title="b5">c</font>,<font color="Olive" title="b3">f</font>,<font color="Olive" title="b4">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">q</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">ic</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Olive" title="b3">f</font>;<br/>
<a NAME="E1:23"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:23_1"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">and </span><br/><a NAME="E2:23_1"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">q</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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">ic</font>, <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">ic</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Olive" title="b3">k</font> &amp; <font color="Olive" title="b2">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> for <font color="Olive" title="b4">c</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Olive" title="b3">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a>   st <font color="Olive" title="b4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Olive" title="b1">ic</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Olive" title="b3">k</font>) holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Olive" title="b4">c</font>,<font color="Olive" title="b2">f</font>,<font color="Olive" title="b3">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">ic</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Olive" title="b2">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">ic</font>, <font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">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"> ( <font color="Maroon" title="c3">ic</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c5">k</font> &amp; <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  for <font color="Olive" title="b1">c</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c5">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c3">ic</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>) holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Olive" title="b1">c</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">ic</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:23_1"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
<font color="Maroon" title="c3">ic</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c5">k</font>
 <span class="kw">and </span><br/><a NAME="E4:23_1"/><span class="lab"><font color="Green" title="E21">A5</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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">c</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c5">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c3">ic</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>) holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Olive" title="b1">c</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">ic</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">c</font> be   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c5">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"> ( <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c3">ic</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>) implies <span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">ic</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font> )</span><br/>

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <a href="mesfunc1.html#K2" title="MESFUNC1:func.2">K560</a>() =  <a href="radix_1.html#K5" title="RADIX_1:func.5">DigB</a> (<font color="Maroon" title="c6">c</font>,<span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>);<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">cn</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <span class="kw"> such that </span><br/><a NAME="E6:23_1"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">cn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E7:23_1"/><span class="lab"><font color="Green" title="E23">A7</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</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> <font color="Maroon" title="c7">cn</font> holds <br/><font color="Maroon" title="c7">cn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></span>
<a NAME="E8:23_1"/><span class="lab"><font color="Green" title="E24">A8</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">cn</font> <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23_1"><span class="lab"><font color="Green" title="E22">A6</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="E9:23_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c7">cn</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c7">cn</font> or <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:23_1_1"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c7">cn</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">xx</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:23_1_1"/><span class="lab"><font color="Green" title="E25">A9</font></span>: 
<font color="Maroon" title="c9">xx</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> <font color="Maroon" title="c7">cn</font>
 <span class="kw">and </span><br/><a NAME="E4:23_1_1"/><span class="lab"><font color="Green" title="E26">A10</font></span>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">cn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">xx</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">xx</font> = <font color="Maroon" title="c9">xx</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a></span>;<br/>
<a NAME="E6:23_1_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">cn</font> <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23_1"><span class="lab"><font color="Green" title="E22">A6</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="E7:23_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">xx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="ref" href="radix_2.html#T5" target="_self" title="RADIX_2:th.5">Th5</a></span>;<br/>
<a NAME="E8:23_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A11</font></span>: 
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<font color="Maroon" title="c6">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">xx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_1.html#T16" title="RADIX_1:th.16">RADIX_1:16</a></span>;<br/>
<font color="Maroon" title="c8">z</font> = 
 <a href="radix_1.html#K5" title="RADIX_1:func.5">DigB</a> (<font color="Maroon" title="c6">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">xx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E3:23_1_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="txt" href="#E4:23_1_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a></span>
<br/>.= 
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<font color="Maroon" title="c6">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">xx</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 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="kw">hence </span><a NAME="E10:23_1_1"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:23_1_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">cn</font> = <font color="Maroon" title="c7">cn</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c5">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">cn</font> = <font color="Maroon" title="c8">cn</font> as   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<a NAME="E12:23_1"/><span class="lab"><font color="Green" title="E25">A12</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">c2</font> =  <a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>);<br/>
<a NAME="E13:23_1"/><span class="lab"><font color="Green" title="E26">A13</font></span>: 
 <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_2.html#T6" target="_self" title="RADIX_2:th.6">Th6</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <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>  = <span class="p1">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">cn2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E15:23_1"/><span class="lab"><font color="Green" title="E27">A14</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">cn2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E16:23_1"/><span class="lab"><font color="Green" title="E28">A15</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</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> <font color="Maroon" title="c11">cn2</font> holds <br/><font color="Maroon" title="c11">cn2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">i</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></span>
<a NAME="E17:23_1"/><span class="lab"><font color="Green" title="E29">A16</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">cn2</font> <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:23_1"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">cn2</font> = <font color="Maroon" title="c11">cn2</font> as   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c1">n</font>, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:23_1"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">icn2</font> =  <a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>);<br/>
<a NAME="E19:23_1"/><span class="lab"><font color="Green" title="E30">A17</font></span>: 
<font color="Maroon" title="c3">ic</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="radix_1.html#D12" title="RADIX_1:def.12">RADIX_1:def 12</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">icn</font> =  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Maroon" title="c9">cn</font>;<br/>
<span class="kw">assume </span><a NAME="E20:23_1"/><span class="lab"><font color="Green" title="E31">A18</font></span>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c3">ic</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">ic</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font></span><br/>

<a NAME="E21:23_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A19</font></span>: 
 <a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_2.html#T14" target="_self" title="RADIX_2:th.14">Th14</a></span>;<br/>
<a NAME="E22:23_1"/><span class="lab"><font color="Green" title="E33">A20</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">cn</font> holds <br/><font color="Maroon" title="c9">cn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">cn2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">i</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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i</font> &amp; <font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">cn</font> implies <font color="Maroon" title="c9">cn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">cn2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:23_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i</font> &amp; <font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">cn</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">cn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">cn2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">i</font></span><br/>

<a NAME="E2:23_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A21</font></span>: 
<font color="Maroon" title="c15">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E3:23_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A22</font></span>: 
<font color="Maroon" title="c15">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_2.html#T5" target="_self" title="RADIX_2:th.5">Th5</a></span>;<br/>
<font color="Maroon" title="c9">cn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i</font> = 
 <a href="radix_1.html#K5" title="RADIX_1:func.5">DigB</a> (<font color="Maroon" title="c6">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E8:23_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E2:23_1_2"><span class="lab"><font color="Green" title="E34">A21</font></span></a></span>
<br/>.= 
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<font color="Maroon" title="c6">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 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/>.= 
<font color="Maroon" title="c6">c</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_2"><span class="lab"><font color="Green" title="E35">A22</font></span></a>, <a class="ref" href="radix_1.html#D3" title="RADIX_1:def.3">RADIX_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:23_1"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="ref" href="radix_2.html#T14" target="_self" title="RADIX_2:th.14">Th14</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:23_1_2"/>
<font color="Maroon" title="c9">cn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">cn2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:23_1"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E17:23_1"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="txt" href="#E2:23_1_2"><span class="lab"><font color="Green" title="E34">A21</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E23:23_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A23</font></span>: 
 <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Maroon" title="c9">cn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E15:23_1"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="ref" href="radix_2.html#T13" target="_self" title="RADIX_2:th.13">Th13</a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>;<br/>
<a NAME="E24:23_1"/>
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#T2" title="FINSEQ_1:th.2">FINSEQ_1:2</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E25:23_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A24</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> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T8" title="FINSEQ_2:th.8">FINSEQ_2:8</a></span>;<br/>
<a NAME="E26:23_1"/><span class="lab"><font color="Green" title="E36">A25</font></span>: 
<font color="Maroon" title="c3">ic</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c5">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:23_1_3"/><span class="lab"><font color="Green" title="E37">A26</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> (<span class="p3">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,1,<font color="Maroon" title="c5">k</font>)</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></span>;<br/>
<a NAME="E2:23_1_3"/><span class="lab"><font color="Green" title="E38">A27</font></span>: 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>3</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <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>  = <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> $1;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">r2</font> =  <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">rD</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E5:23_1_3"/><span class="lab"><font color="Green" title="E39">A28</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">rD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E6:23_1_3"/><span class="lab"><font color="Green" title="E40">A29</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</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> <font color="Maroon" title="c16">rD</font> holds <br/><font color="Maroon" title="c16">rD</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">i</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c17">rD</font> = <font color="Maroon" title="c16">rD</font> as   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c1">n</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23_1_3"><span class="lab"><font color="Green" title="E39">A28</font></span></a>, <a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<span class="lab"><font color="Green" title="E41">A30</font></span>:  <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="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
<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/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">rD1</font> = <font color="Maroon" title="c17">rD</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T131" title="FINSEQ_2:th.131">FINSEQ_2:131</a></span>;<br/>
<a NAME="E10:23_1_3"/><span class="lab"><font color="Green" title="E42">A31</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">rD</font> <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23_1_3"><span class="lab"><font color="Green" title="E39">A28</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E43">A32</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <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="c5">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">r2</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c18">rD1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23_1_3"><span class="lab"><font color="Green" title="E40">A29</font></span></a>, <a class="txt" href="#E8:23_1_3"><span class="lab"><font color="Green" title="E41">A30</font></span></a>, <a class="ref" href="finseq_1.html#T13" title="FINSEQ_1:th.13">FINSEQ_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<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/>
<span class="lab"><font color="Green" title="E44">A33</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> (<span class="p4">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,1,<font color="Maroon" title="c5">k</font>)</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> (<span class="p4">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,1,<font color="Maroon" title="c5">k</font>)</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:23_1_3"><span class="lab"><font color="Green" title="E43">A32</font></span></a>, <a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>
;<br/>
<a NAME="E13:23_1_3"/><span class="lab"><font color="Green" title="E45">A34</font></span>: 
 <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> (<span class="p2">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.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="E14:23_1_3"/><span class="lab"><font color="Green" title="E46">A35</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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> (<span class="p2">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<span class="p2">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> (<span class="p4">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,1,<font color="Maroon" title="c5">k</font>)</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_3_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">i</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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i</font> &amp; <font color="Maroon" title="c19">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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> (<span class="p2">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> implies <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<span class="p2">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> (<span class="p4">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,1,<font color="Maroon" title="c5">k</font>)</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:23_1_3_4"/><span class="lab"><font color="Green" title="E47">A36</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i</font>
 <span class="kw">and </span><br/><a NAME="E2:23_1_3_4"/><span class="lab"><font color="Green" title="E48">A37</font></span>: 
<font color="Maroon" title="c19">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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> (<span class="p2">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<span class="p2">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> (<span class="p4">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,1,<font color="Maroon" title="c5">k</font>)</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font></span><br/>

<a NAME="E3:23_1_3_4"/><span class="lab"><font color="Green" title="E49">A38</font></span>: 
<font color="Maroon" title="c19">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1_3_4"><span class="lab"><font color="Green" title="E48">A37</font></span></a>, <a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<a NAME="E4:23_1_3_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A39</font></span>: 
<font color="Maroon" title="c19">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_3_4"><span class="lab"><font color="Green" title="E47">A36</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E5:23_1_3_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A40</font></span>: 
<font color="Maroon" title="c19">i</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> (<span class="p2">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:23_1_3"><span class="lab"><font color="Green" title="E45">A34</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_3_4_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:23_1_3_4_1"/>
( <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c19">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_3_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:23_1_3_4_1_1"/><span class="lab"><font color="Green" title="E52">A41</font></span>: 
<font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<span class="p2">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> (<span class="p4">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,1,<font color="Maroon" title="c5">k</font>)</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> (<span class="p4">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,1,<font color="Maroon" title="c5">k</font>)</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> = 
 <a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> (<span class="p1">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,1,<font color="Maroon" title="c5">k</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T41" title="FINSEQ_1:th.41">FINSEQ_1:41</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<span class="p2">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:23_1"><span class="lab"><font color="Green" title="E35">A24</font></span></a>, <a class="ref" href="radix_2.html#D2" target="_self" title="RADIX_2:def.2">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<span class="p2">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23_1_3_4"><span class="lab"><font color="Green" title="E51">A40</font></span></a>, <a class="txt" href="#E1:23_1_3_4_1_1"><span class="lab"><font color="Green" title="E52">A41</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:23_1_3_4_1_1"/>
<span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<span class="p2">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> (<span class="p4">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,1,<font color="Maroon" title="c5">k</font>)</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_3_4_1_1"><span class="lab"><font color="Green" title="E52">A41</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_3_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:23_1_3_4_1_2"/><span class="lab"><font color="Green" title="E52">A42</font></span>: 
<font color="Maroon" title="c19">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<span class="p2">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> (<span class="p4">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,1,<font color="Maroon" title="c5">k</font>)</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c20">rs2</font> = <span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> as    <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>  ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">rs</font> = <font color="Maroon" title="c17">rD</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E4:23_1_3_4_1_2"/>
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_3_4"><span class="lab"><font color="Green" title="E47">A36</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">i1</font> = <font color="Maroon" title="c19">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/><a NAME="E6:23_1_3_4_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_3_4"><span class="lab"><font color="Green" title="E47">A36</font></span></a>, <a class="txt" href="#E1:23_1_3_4_1_2"><span class="lab"><font color="Green" title="E52">A42</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E7:23_1_3_4_1_2"/><span class="kw">then </span>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T7" title="INT_1:th.7">INT_1:7</a></span>;<br/><a NAME="E8:23_1_3_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A43</font></span>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E9:23_1_3_4_1_2"/>
<font color="Maroon" title="c19">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_3_4"><span class="lab"><font color="Green" title="E49">A38</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E10:23_1_3_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A44</font></span>: 
<font color="Maroon" title="c22">i1</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:23_1_3_4_1_2"><span class="lab"><font color="Green" title="E53">A43</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E11:23_1_3_4_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_3_4"><span class="lab"><font color="Green" title="E47">A36</font></span></a>, <a class="txt" href="#E1:23_1_3_4_1_2"><span class="lab"><font color="Green" title="E52">A42</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> (<span class="p4">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,1,<font color="Maroon" title="c5">k</font>)</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> = 
<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="c5">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_3"><span class="lab"><font color="Green" title="E37">A26</font></span></a>, <a class="txt" href="#E12:23_1_3"><span class="lab"><font color="Green" title="E44">A33</font></span></a>, <a class="txt" href="#E3:23_1_3_4"><span class="lab"><font color="Green" title="E49">A38</font></span></a>, <a class="ref" href="finseq_1.html#T24" title="FINSEQ_1:th.24">FINSEQ_1:24</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">r2</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c17">rD</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23_1_3"><span class="lab"><font color="Green" title="E40">A29</font></span></a>, <a class="txt" href="#E8:23_1_3"><span class="lab"><font color="Green" title="E41">A30</font></span></a>, <a class="txt" href="#E10:23_1_3"><span class="lab"><font color="Green" title="E42">A31</font></span></a>, <a class="ref" href="finseq_1.html#T13" title="FINSEQ_1:th.13">FINSEQ_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c15">r2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c21">rs</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T45" title="RVSUM_1:th.45">RVSUM_1:45</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">rs2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23_1_3"><span class="lab"><font color="Green" title="E40">A29</font></span></a>, <a class="txt" href="#E8:23_1_3"><span class="lab"><font color="Green" title="E41">A30</font></span></a>, <a class="txt" href="#E10:23_1_3"><span class="lab"><font color="Green" title="E42">A31</font></span></a>, <a class="ref" href="finseq_1.html#T13" title="FINSEQ_1:th.13">FINSEQ_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">i1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:23_1_3"><span class="lab"><font color="Green" title="E41">A30</font></span></a>, <a class="txt" href="#E10:23_1_3_4_1_2"><span class="lab"><font color="Green" title="E54">A44</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c22">i1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:23_1_3_4_1_2"><span class="lab"><font color="Green" title="E54">A44</font></span></a>, <a class="ref" href="radix_2.html#D2" target="_self" title="RADIX_2:def.2">Def2</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="c5">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">cn2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i1</font></span>)</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="c5">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c22">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">cn2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T6" title="NEWTON:th.6">NEWTON: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="c5">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c22">i1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">cn2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:23_1_3_4_1_2"><span class="lab"><font color="Green" title="E53">A43</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</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="c5">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c22">i1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:23_1"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E17:23_1"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="txt" href="#E10:23_1_3_4_1_2"><span class="lab"><font color="Green" title="E54">A44</font></span></a></span>
<br/>.= 
 <a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> (<span class="p1">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c19">i</font>,<font color="Maroon" title="c5">k</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_3_4"><span class="lab"><font color="Green" title="E47">A36</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<span class="p2">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1_3_4"><span class="lab"><font color="Green" title="E50">A39</font></span></a>, <a class="ref" href="radix_2.html#D2" target="_self" title="RADIX_2:def.2">Def2</a></span>
;<br/><span class="kw">hence </span><a NAME="E13:23_1_3_4_1_2"/>
<span class="p1">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<span class="p2">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> (<span class="p4">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,1,<font color="Maroon" title="c5">k</font>)</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23_1_3_4"><span class="lab"><font color="Green" title="E51">A40</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E15:23_1_3"/>
 <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> (<span class="p2">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> (<span class="p4">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,1,<font color="Maroon" title="c5">k</font>)</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:23_1_3"><span class="lab"><font color="Green" title="E44">A33</font></span></a>, <a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<a NAME="E16:23_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A45</font></span>: 
 <a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<span class="p1">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c5">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> (<span class="p3">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,1,<font color="Maroon" title="c5">k</font>)</span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <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="c5">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:23_1_3"><span class="lab"><font color="Green" title="E46">A35</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>;<br/>
<font color="Maroon" title="c3">ic</font> = 
 <a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<span class="p1">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c5">k</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="radix_2.html#T15" target="_self" title="RADIX_2:th.15">Th15</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> (<span class="p2">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,1,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K2" title="RADIX_2:func.2">DigitSD2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:23_1_3"><span class="lab"><font color="Green" title="E47">A45</font></span></a>, <a class="ref" href="rvsum_1.html#T76" title="RVSUM_1:th.76">RVSUM_1:76</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> (<span class="p2">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,1,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">r2</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c17">rD</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23_1_3"><span class="lab"><font color="Green" title="E40">A29</font></span></a>, <a class="txt" href="#E8:23_1_3"><span class="lab"><font color="Green" title="E41">A30</font></span></a>, <a class="txt" href="#E10:23_1_3"><span class="lab"><font color="Green" title="E42">A31</font></span></a>, <a class="ref" href="finseq_1.html#T13" title="FINSEQ_1:th.13">FINSEQ_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> (<span class="p2">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,1,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">r2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c17">rD</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T87" title="RVSUM_1:th.87">RVSUM_1:87</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K1" title="RADIX_2:func.1">SubDigit2</a> (<span class="p2">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span>,1,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <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="c5">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23_1_3"><span class="lab"><font color="Green" title="E40">A29</font></span></a>, <a class="txt" href="#E8:23_1_3"><span class="lab"><font color="Green" title="E41">A30</font></span></a>, <a class="txt" href="#E10:23_1_3"><span class="lab"><font color="Green" title="E42">A31</font></span></a>, <a class="ref" href="finseq_1.html#T13" title="FINSEQ_1:th.13">FINSEQ_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">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="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <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="c5">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1_3"><span class="lab"><font color="Green" title="E38">A27</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <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="c5">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></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/>
<span class="kw">hence </span><a NAME="E18:23_1_3"/>
<font color="Maroon" title="c3">ic</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c5">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="lab"><font color="Green" title="E37">A46</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">ic</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font>

<br/>.= 
<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" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p5">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T66" title="NAT_D:th.66">NAT_D:66</a></span>
<br/>.= 
<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" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p0">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="radix_2.html#T3" target="_self" title="RADIX_2:th.3">Th3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p5">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="radix_2.html#T2" target="_self" title="RADIX_2:th.2">Th2</a></span>
;<br/>
<a NAME="E28:23_1"/><span class="lab"><font color="Green" title="E38">A47</font></span>: 
<font color="Maroon" title="c9">cn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">cn2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E15:23_1"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="txt" href="#E22:23_1"><span class="lab"><font color="Green" title="E33">A20</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>;<br/>
<a NAME="E29:23_1"/><span class="lab"><font color="Green" title="E39">A48</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">cn</font> holds <br/><font color="Maroon" title="c9">cn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<span class="p2">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_5"><span class="kw">proof </span></a><div class="add">

<span class="lab"><font color="Green" title="E40">A49</font></span>: <span class="p1">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 = 
 <a href="radix_2.html#K4" title="RADIX_2:func.4">DigitDC2</a> (<font color="Maroon" title="c3">ic</font>,1,<font color="Maroon" title="c5">k</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:23_1"><span class="lab"><font color="Green" title="E35">A24</font></span></a>, <a class="ref" href="radix_2.html#D5" target="_self" title="RADIX_2:def.5">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">ic</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <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="c5">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>
<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/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">ic</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T4" title="NEWTON:th.4">NEWTON:4</a></span>
<br/>.= 
<font color="Maroon" title="c3">ic</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <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="c5">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T4" title="NAT_2:th.4">NAT_2:4</a></span>
<br/>.= 
<font color="Maroon" title="c3">ic</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T5" title="NEWTON:th.5">NEWTON:5</a></span>
;<br/>
<a NAME="E2:23_1_5"/><span class="lab"><font color="Green" title="E41">A50</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_2.html#T6" target="_self" title="RADIX_2:th.6">Th6</a></span>;<br/>
<span class="lab"><font color="Green" title="E42">A51</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> = 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p5">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></span>
<br/>.= 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1_5"><span class="lab"><font color="Green" title="E41">A50</font></span></a>, <a class="ref" href="xcmplx_1.html#T113" title="XCMPLX_1:th.113">XCMPLX_1:113</a></span>
<br/>.= 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T28" title="INT_1:th.28">INT_1:28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></span>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:23_1"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="txt" href="#E1:23_1_5"><span class="lab"><font color="Green" title="E40">A49</font></span></a>, <a class="ref" href="nat_d.html#T1" title="NAT_D:th.1">NAT_D:1</a>, <a class="ref" href="nat_d.html#T27" title="NAT_D:th.27">NAT_D:27</a></span>
;<br/>
<a NAME="E4:23_1_5"/><span class="lab"><font color="Green" title="E43">A52</font></span>: 
 <a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_2.html#T6" target="_self" title="RADIX_2:th.6">Th6</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c15">i</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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i</font> &amp; <font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">cn</font> implies <font color="Maroon" title="c9">cn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<span class="p2">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">i</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E5:23_1_5"/><span class="lab"><font color="Green" title="E44">A53</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i</font>
 <span class="kw">and </span><br/><a NAME="E6:23_1_5"/><span class="lab"><font color="Green" title="E45">A54</font></span>: 
<font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">cn</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">cn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<span class="p2">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">i</font></span><br/>

<a NAME="E7:23_1_5"/><span class="lab"><font color="Green" title="E46">A55</font></span>: 
<font color="Maroon" title="c15">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E5:23_1_5"><span class="lab"><font color="Green" title="E44">A53</font></span></a>, <a class="txt" href="#E6:23_1_5"><span class="lab"><font color="Green" title="E45">A54</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">i</font> = <font color="Maroon" title="c15">i</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E9:23_1_5"/><span class="lab"><font color="Green" title="E47">A56</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23_1_5"><span class="lab"><font color="Green" title="E44">A53</font></span></a>, <a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/>
<a NAME="E10:23_1_5"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E5:23_1_5"><span class="lab"><font color="Green" title="E44">A53</font></span></a>, <a class="txt" href="#E6:23_1_5"><span class="lab"><font color="Green" title="E45">A54</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a>, <a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/>
<a NAME="E11:23_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A57</font></span>: 
<font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<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/>
<font color="Maroon" title="c9">cn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">i</font> = 
<span class="p1">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:23_1"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E17:23_1"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="txt" href="#E28:23_1"><span class="lab"><font color="Green" title="E38">A47</font></span></a>, <a class="txt" href="#E7:23_1_5"><span class="lab"><font color="Green" title="E46">A55</font></span></a></span>
<br/>.= 
 <a href="radix_2.html#K4" title="RADIX_2:func.4">DigitDC2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:23_1_5"><span class="lab"><font color="Green" title="E48">A57</font></span></a>, <a class="ref" href="radix_2.html#D5" target="_self" title="RADIX_2:def.5">Def5</a></span>
<br/>.= 
<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_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:23_1"><span class="lab"><font color="Green" title="E36">A25</font></span></a>, <a class="txt" href="#E9:23_1_5"><span class="lab"><font color="Green" title="E47">A56</font></span></a>, <a class="ref" href="radix_2.html#T4" target="_self" title="RADIX_2:th.4">Th4</a>, <a class="ref" href="radix_2.html#T6" target="_self" title="RADIX_2:th.6">Th6</a></span>
<br/>.= 
<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_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>
<br/>.= 
<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_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23_1_5"><span class="lab"><font color="Green" title="E44">A53</font></span></a>, <a class="txt" href="#E4:23_1_5"><span class="lab"><font color="Green" title="E43">A52</font></span></a>, <a class="ref" href="pepin.html#T26" title="PEPIN:th.26">PEPIN:26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_5"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="ref" href="nat_2.html#T27" title="NAT_2:th.27">NAT_2:27</a></span>
<br/>.= 
 <a href="radix_2.html#K4" title="RADIX_2:func.4">DigitDC2</a> (<span class="p1">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c5">k</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23_1_5"><span class="lab"><font color="Green" title="E44">A53</font></span></a>, <a class="ref" href="radix_2.html#T4" target="_self" title="RADIX_2:th.4">Th4</a>, <a class="ref" href="radix_2.html#T6" target="_self" title="RADIX_2:th.6">Th6</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E13:23_1_5"/>
<font color="Maroon" title="c9">cn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<span class="p2">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23_1_5"><span class="lab"><font color="Green" title="E46">A55</font></span></a>, <a class="ref" href="radix_2.html#D5" target="_self" title="RADIX_2:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c15">icn</font> =  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Maroon" title="c9">cn</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:23_1"><span class="lab"><font color="Green" title="E34">A23</font></span></a></span>;<br/>
<a NAME="E31:23_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">cn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<span class="p2">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<a NAME="E32:23_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">cn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<span class="p1">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">k</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:23_1"><span class="lab"><font color="Green" title="E39">A48</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>;<br/>
<a NAME="E33:23_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A58</font></span>: 
<font color="Maroon" title="c9">cn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c15">icn</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">k</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:23_1"><span class="lab"><font color="Green" title="E34">A23</font></span></a>, <a class="ref" href="radix_2.html#T14" target="_self" title="RADIX_2:th.14">Th14</a></span>;<br/>
<a NAME="E34:23_1"/>
<font color="Maroon" title="c3">ic</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:23_1"><span class="lab"><font color="Green" title="E36">A25</font></span></a>, <a class="ref" href="int_1.html#T6" title="INT_1:th.6">INT_1:6</a></span>;<br/>
<a NAME="E35:23_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:23_1"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E36:23_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <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="c5">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T6" title="NEWTON:th.6">NEWTON:6</a></span>;<br/>
<a NAME="E37:23_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">icn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:23_1"><span class="lab"><font color="Green" title="E34">A23</font></span></a>, <a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/>
<a NAME="E38:23_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A59</font></span>: 
<font color="Maroon" title="c15">icn</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_1.html#D12" title="RADIX_1:def.12">RADIX_1:def 12</a></span>;<br/>
<a NAME="E39:23_1"/><span class="lab"><font color="Green" title="E42">A60</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <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>   st <font color="Olive" title="b1">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> holds <br/> <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<font color="Maroon" title="c9">cn</font>,<font color="Olive" title="b1">i</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="c6">c</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">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  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<font color="Maroon" title="c9">cn</font>,<font color="Maroon" title="c16">i</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="c6">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:23_1_6"/><span class="lab"><font color="Green" title="E43">A61</font></span>: 
<font color="Maroon" title="c16">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<font color="Maroon" title="c9">cn</font>,<font color="Maroon" title="c16">i</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="c6">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span><br/>

 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<font color="Maroon" title="c6">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) = 
 <a href="radix_1.html#K5" title="RADIX_1:func.5">DigB</a> (<font color="Maroon" title="c6">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 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/>.= 
<font color="Maroon" title="c9">cn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E8:23_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E1:23_1_6"><span class="lab"><font color="Green" title="E43">A61</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:23_1_6"/>
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<font color="Maroon" title="c9">cn</font>,<font color="Maroon" title="c16">i</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="c6">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_6"><span class="lab"><font color="Green" title="E43">A61</font></span></a>, <a class="ref" href="radix_1.html#D3" title="RADIX_1:def.3">RADIX_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E40:23_1"/><span class="lab"><font color="Green" title="E43">A62</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <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>   st <font color="Olive" title="b1">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> holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c9">cn</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <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> ] <span class="kw">means </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> <font color="Maroon" title="c1">n</font> implies <span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c9">cn</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1 );<br/>
<a NAME="E1:23_1_7"/><span class="lab"><font color="Green" title="E44">A63</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <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>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_7_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c16">i</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:23_1_7_1"/><span class="lab"><font color="Green" title="E45">A64</font></span>: 
( <font color="Maroon" title="c16">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 <span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c9">cn</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">i</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<a NAME="E2:23_1_7_1"/><span class="lab"><font color="Green" title="E46">A65</font></span>: 
( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:23_1_7_1"/>
<font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 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> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c9">cn</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/>

<a NAME="E4:23_1_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A66</font></span>: 
<font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<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/>
<a NAME="E5:23_1_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A67</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E6:23_1_7_1"/><span class="lab"><font color="Green" title="E49">A68</font></span>: 
<font color="Maroon" title="c1">n</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_7_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_7_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:23_1_7_1_1_1"/>
( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1_7_1"><span class="lab"><font color="Green" title="E46">A65</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_7_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:23_1_7_1_1_1_1"/><span class="lab"><font color="Green" title="E50">A69</font></span>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c9">cn</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c9">cn</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
 <a href="radix_2.html#K6" title="RADIX_2:func.6">Table1</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c9">cn</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="radix_2.html#D7" target="_self" title="RADIX_2:def.7">Def7</a></span>
<br/>.= 
 <a href="radix_2.html#K6" title="RADIX_2:func.6">Table1</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:23_1"><span class="lab"><font color="Green" title="E42">A60</font></span></a>, <a class="txt" href="#E6:23_1_7_1"><span class="lab"><font color="Green" title="E49">A68</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:23_1"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="ref" href="radix_2.html#D7" target="_self" title="RADIX_2:def.7">Def7</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:23_1_7_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c9">cn</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_7_1_1_1_1"><span class="lab"><font color="Green" title="E50">A69</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_7_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:23_1_7_1_1_1_2"/><span class="lab"><font color="Green" title="E50">A70</font></span>: 
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c9">cn</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c17">nn</font> = <font color="Maroon" title="c1">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as    <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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="int_1.html#T5" title="INT_1:th.5">INT_1:5</a></span>;<br/><a NAME="E3:23_1_7_1_1_1_2"/><span class="lab"><font color="Green" title="E51">A71</font></span>: 
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">nn</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23_1_7_1"><span class="lab"><font color="Green" title="E48">A67</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E4:23_1_7_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 ;<br/><a NAME="E5:23_1_7_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A72</font></span>: 
 ex <font color="Olive" title="b1">I1</font>, <font color="Olive" title="b2">I2</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Olive" title="b1">I1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Olive" title="b2">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">I1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="radix_2.html#K6" title="RADIX_2:func.6">Table1</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c16">i</font></span>)</span>)</span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:23_1"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="txt" href="#E1:23_1_7_1_1_1_2"><span class="lab"><font color="Green" title="E50">A70</font></span></a>, <a class="ref" href="radix_2.html#D7" target="_self" title="RADIX_2:def.7">Def7</a></span>;<br/><a NAME="E6:23_1_7_1_1_1_2"/><span class="lab"><font color="Green" title="E53">A73</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T35" title="NAT_D:th.35">NAT_D:35</a></span>;<br/><a NAME="E7:23_1_7_1_1_1_2"/>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c16">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1_7_1"><span class="lab"><font color="Green" title="E47">A66</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E8:23_1_7_1_1_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c16">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_7_1_1_1_2"><span class="lab"><font color="Green" title="E51">A71</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><a NAME="E9:23_1_7_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A74</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c16">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23_1_7_1_1_1_2"><span class="lab"><font color="Green" title="E53">A73</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E10:23_1_7_1_1_1_2"/>
 ex <font color="Olive" title="b1">I1</font>, <font color="Olive" title="b2">I2</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Olive" title="b1">I1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c9">cn</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Olive" title="b2">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c9">cn</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">I1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="radix_2.html#K6" title="RADIX_2:func.6">Table1</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c9">cn</font>,<font color="Maroon" title="c4">f</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c16">i</font></span>)</span>)</span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E5:23_1_7_1"><span class="lab"><font color="Green" title="E48">A67</font></span></a>, <a class="txt" href="#E1:23_1_7_1_1_1_2"><span class="lab"><font color="Green" title="E50">A70</font></span></a>, <a class="ref" href="radix_2.html#D7" target="_self" title="RADIX_2:def.7">Def7</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c9">cn</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<font color="Maroon" title="c6">c</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:23_1"><span class="lab"><font color="Green" title="E42">A60</font></span></a>, <a class="txt" href="#E1:23_1_7_1"><span class="lab"><font color="Green" title="E45">A64</font></span></a>, <a class="txt" href="#E1:23_1_7_1_1_1_2"><span class="lab"><font color="Green" title="E50">A70</font></span></a>, <a class="txt" href="#E3:23_1_7_1_1_1_2"><span class="lab"><font color="Green" title="E51">A71</font></span></a>, <a class="txt" href="#E9:23_1_7_1_1_1_2"><span class="lab"><font color="Green" title="E54">A74</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_7_1_1_1_2"><span class="lab"><font color="Green" title="E51">A71</font></span></a>, <a class="txt" href="#E5:23_1_7_1_1_1_2"><span class="lab"><font color="Green" title="E52">A72</font></span></a>, <a class="ref" href="nat_d.html#T38" title="NAT_D:th.38">NAT_D:38</a></span>
;<br/><span class="kw">hence </span><a NAME="E12:23_1_7_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c9">cn</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:23_1_7_1"/>
<span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c9">cn</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:23_1_7"/><span class="lab"><font color="Green" title="E45">A75</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<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/>
<a NAME="E3:23_1_7"/>
 for <font color="Olive" title="b1">i</font> being    <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>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E2:23_1_7"><span class="lab"><font color="Green" title="E45">A75</font></span></a>, <a class="txt" href="#E1:23_1_7"><span class="lab"><font color="Green" title="E44">A63</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:23_1_7"/>
 for <font color="Olive" title="b1">i</font> being    <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>   st <font color="Olive" title="b1">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> holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c9">cn</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E41:23_1"/>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 &amp; <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E42:23_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A76</font></span>: 
 ex <font color="Olive" title="b1">I1</font>, <font color="Olive" title="b2">I2</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Olive" title="b1">I1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b2">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">I1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="radix_2.html#K6" title="RADIX_2:func.6">Table1</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="radix_2.html#D7" target="_self" title="RADIX_2:def.7">Def7</a></span>;<br/>
<a NAME="E43:23_1"/>
<font color="Maroon" title="c1">n</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c9">cn</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="radix_2.html#K6" title="RADIX_2:func.6">Table1</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:23_1"><span class="lab"><font color="Green" title="E43">A62</font></span></a>, <a class="txt" href="#E42:23_1"><span class="lab"><font color="Green" title="E44">A76</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c15">icn</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="radix_2.html#K6" title="RADIX_2:func.6">Table1</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E4:23_1"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E38:23_1"><span class="lab"><font color="Green" title="E41">A59</font></span></a>, <a class="txt" href="#E33:23_1"><span class="lab"><font color="Green" title="E40">A58</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p5">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="radix_2.html#K6" title="RADIX_2:func.6">Table1</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E15:23_1"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="txt" href="#E22:23_1"><span class="lab"><font color="Green" title="E33">A20</font></span></a>, <a class="ref" href="radix_2.html#T13" target="_self" title="RADIX_2:th.13">Th13</a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p5">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> (<font color="Maroon" title="c6">c</font>,1)</span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p5">(<span class="default"><a href="radix_2.html#K3" title="RADIX_2:func.3">SDDec2</a> (<font color="Maroon" title="c12">cn2</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="radix_2.html#K5" title="RADIX_2:func.5">DecSD2</a> (<font color="Maroon" title="c3">ic</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:23_1"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="txt" href="#E25:23_1"><span class="lab"><font color="Green" title="E35">A24</font></span></a>, <a class="ref" href="radix_1.html#D3" title="RADIX_1:def.3">RADIX_1:def 3</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E45:23_1"/>
<span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">ic</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:23_1"><span class="lab"><font color="Green" title="E37">A46</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:23"/><span class="lab"><font color="Green" title="E18">A77</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">q</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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">ic</font>, <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">ic</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> 1,<font color="Olive" title="b3">k</font> &amp; <font color="Olive" title="b2">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> for <font color="Olive" title="b4">c</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of 1,<font color="Olive" title="b3">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a>   st <font color="Olive" title="b4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Olive" title="b1">ic</font>,1,<font color="Olive" title="b3">k</font>) holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c1">q</font>,<font color="Olive" title="b4">c</font>,<font color="Olive" title="b2">f</font>,<font color="Olive" title="b3">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">ic</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Olive" title="b2">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">ic</font>, <font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">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"> ( <font color="Maroon" title="c2">ic</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> 1,<font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  for <font color="Olive" title="b1">c</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of 1,<font color="Maroon" title="c4">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c2">ic</font>,1,<font color="Maroon" title="c4">k</font>) holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c1">q</font>,<font color="Olive" title="b1">c</font>,<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">ic</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">f</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:23_2"/><span class="lab"><font color="Green" title="E19">A78</font></span>: 
<font color="Maroon" title="c2">ic</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> 1,<font color="Maroon" title="c4">k</font>
 <span class="kw">and </span><br/><a NAME="E2:23_2"/>
<font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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">c</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of 1,<font color="Maroon" title="c4">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c2">ic</font>,1,<font color="Maroon" title="c4">k</font>) holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c1">q</font>,<font color="Olive" title="b1">c</font>,<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">ic</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">f</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">c</font> be   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of 1,<font color="Maroon" title="c4">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"> ( <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c2">ic</font>,1,<font color="Maroon" title="c4">k</font>) implies <span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c1">q</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">ic</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:23_2"/><span class="lab"><font color="Green" title="E20">A79</font></span>: 
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c2">ic</font>,1,<font color="Maroon" title="c4">k</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c1">q</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">ic</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">f</font></span><br/>

<span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c1">q</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 = 
 <a href="radix_2.html#K6" title="RADIX_2:func.6">Table1</a> (<font color="Maroon" title="c1">q</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">f</font>,1)
<span class="kw">by</span> <span class="lab"><a class="ref" href="radix_2.html#D7" target="_self" title="RADIX_2:def.7">Def7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Maroon" title="c2">ic</font>,1,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_2"><span class="lab"><font color="Green" title="E20">A79</font></span></a>, <a class="ref" href="radix_2.html#T7" target="_self" title="RADIX_2:th.7">Th7</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:23_2"/>
<span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Maroon" title="c1">q</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">ic</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_2"><span class="lab"><font color="Green" title="E19">A78</font></span></a>, <a class="ref" href="radix_1.html#T22" title="RADIX_1:th.22">RADIX_1:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:23"/>
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S8" title="NAT_1:sch.8">NAT_1:sch 8</a>(<a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E18">A77</font></span></a>, <a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E17">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:23"/>
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/> for <font color="Olive" title="b2">q</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><br/>  for <font color="Olive" title="b3">ic</font>, <font color="Olive" title="b4">f</font>, <font color="Olive" title="b5">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">ic</font> <a href="radix_1.html#R1" title="RADIX_1:pred.1">is_represented_by</a> <font color="Olive" title="b1">n</font>,<font color="Olive" title="b5">k</font> &amp; <font color="Olive" title="b4">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> for <font color="Olive" title="b6">c</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Olive" title="b1">n</font>,<font color="Olive" title="b5">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a>   st <font color="Olive" title="b6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K10" title="RADIX_1:func.10">DecSD</a> (<font color="Olive" title="b3">ic</font>,<font color="Olive" title="b1">n</font>,<font color="Olive" title="b5">k</font>) holds <br/><span class="p1">(<span class="default"><a href="radix_2.html#K7" title="RADIX_2:func.7">Mul_mod</a> (<font color="Olive" title="b2">q</font>,<font color="Olive" title="b6">c</font>,<font color="Olive" title="b4">f</font>,<font color="Olive" title="b5">k</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">q</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b3">ic</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Olive" title="b4">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
