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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">k</font> <a href="radix_3.html#K4" title="RADIX_3:func.4">-SD_Sub</a>  =  <a href="radix_3.html#K9" title="RADIX_3:func.9">SD2SDSubDigitS</a> <font color="Maroon" title="c3">x</font>,$1,<font color="Maroon" title="c2">k</font>;<br/>
<b>consider </b><font color="Maroon" title="c4">z</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">k</font> <a href="radix_3.html#K4" title="RADIX_3:func.4">-SD_Sub</a> <b> such that </b><br/><a NAME="E2:32_1_1"/><i><font color="Green" title="E26">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">z</font> <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
 <b>and </b><br/><a NAME="E3:32_1_1"/><i><font color="Green" title="E27">A2</font></i>: 
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">z</font> holds <br/><font color="Maroon" title="c4">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">j</font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></i>
<a NAME="E4:32_1_1"/><i><font color="Green" title="E28">A3</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>
 
<b>by </b><i><a class="txt" href="radix_3.html#E2:32_1_1"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c4">z</font> as   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <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"><font color="Maroon" title="c2">k</font> <a href="radix_3.html#K4" title="RADIX_3:func.4">-SD_Sub</a> </span>)</span> <b>by </b><i><a class="txt" href="radix_3.html#E2:32_1_1"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">z</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">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#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> holds <br/> <a href="radix_3.html#K7" title="RADIX_3:func.7">DigA_SDSub</a> <font color="Maroon" title="c5">z</font>,<font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_3.html#K9" title="RADIX_3:func.9">SD2SDSubDigitS</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">k</font></span><br/>

<b>let </b><font color="Maroon" title="c6">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">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> implies  <a href="radix_3.html#K7" title="RADIX_3:func.7">DigA_SDSub</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_3.html#K9" title="RADIX_3:func.9">SD2SDSubDigitS</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c2">k</font> )</span><br/>

<b>assume </b><a NAME="E6:32_1_1"/><i><font color="Green" title="E29">A4</font></i>: 
<font color="Maroon" title="c6">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="radix_3.html#K7" title="RADIX_3:func.7">DigA_SDSub</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_3.html#K9" title="RADIX_3:func.9">SD2SDSubDigitS</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c2">k</font></span><br/>

<b>hence </b> <a href="radix_3.html#K7" title="RADIX_3:func.7">DigA_SDSub</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c6">i</font> = 
<font color="Maroon" title="c5">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
<b>by </b><i><a class="ref" href="radix_3.html#D5" target="_self" title="RADIX_3:def.5">Def5</a></i>
<br/>.= 
 <a href="radix_3.html#K9" title="RADIX_3:func.9">SD2SDSubDigitS</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c2">k</font>
<b>by </b><i><a class="txt" href="radix_3.html#E3:32_1_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="radix_3.html#E6:32_1_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="radix_3.html#E4:32_1_1"><i><font color="Green" title="E28">A3</font></i></a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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