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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">tx</font>, <font color="Olive" title="b3">ty</font> being  <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of $1,<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a> </span>)</span>  st ( for <font color="Olive" title="b4">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">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> $1 holds <br/> <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Olive" title="b2">tx</font>,<font color="Olive" title="b4">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Olive" title="b3">ty</font>,<font color="Olive" title="b4">i</font> ) holds <br/> <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Olive" title="b2">tx</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Olive" title="b3">ty</font>;<br/>
<a NAME="E1:15"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">tx</font>, <font color="Olive" title="b2">ty</font> being  <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of 1,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a> </span>)</span>  st ( for <font color="Olive" title="b3">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">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> 1 holds <br/> <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Olive" title="b1">tx</font>,<font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Olive" title="b2">ty</font>,<font color="Olive" title="b3">i</font> ) holds <br/> <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Olive" title="b1">tx</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Olive" title="b2">ty</font></span><br/><b>let </b><font color="Maroon" title="c2">tx</font>, <font color="Maroon" title="c3">ty</font> be   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of 1,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a> </span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( 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> 1 holds <br/> <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c2">tx</font>,<font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c3">ty</font>,<font color="Olive" title="b1">i</font> ) implies  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Maroon" title="c2">tx</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Maroon" title="c3">ty</font> )</span><br/>



<b>assume </b><a NAME="E1:15_1"/><i><font color="Green" title="E14">A2</font></i>: 
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> 1 holds <br/> <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c2">tx</font>,<font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c3">ty</font>,<font color="Olive" title="b1">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Maroon" title="c2">tx</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Maroon" title="c3">ty</font></span><br/>

<a NAME="E2:15_1"/><i><font color="Green" title="E15">A3</font></i>: 
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
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:15_1"/><i><font color="Green" title="E16">A4</font></i>: 
 <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Maroon" title="c2">tx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c2">tx</font>,1
 
<b>by </b><i><a class="ref" href="radix_5.html#T4" target="_self" title="RADIX_5:th.4">Th4</a></i>;<br/>
<a NAME="E4:15_1"/>
 <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Maroon" title="c3">ty</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="c3">ty</font>,1
 
<b>by </b><i><a class="ref" href="radix_5.html#T4" target="_self" title="RADIX_5:th.4">Th4</a></i>;<br/>
<b>hence </b><a NAME="E5:15_1"/>
 <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Maroon" title="c2">tx</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Maroon" title="c3">ty</font>
 <b>by </b><i><a class="txt" href="#E1:15_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="#E2:15_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="#E3:15_1"><i><font color="Green" title="E16">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:15"/><i><font color="Green" title="E14">A5</font></i>: 
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="15_2"><b>proof </b></a><div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>assume </b><a NAME="E1:15_2"/><i><font color="Green" title="E15">A6</font></i>: 
( <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>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>let </b><font color="Maroon" title="c2">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">tx</font>, <font color="Olive" title="b2">ty</font> being  <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_1.html#K3" title="RADIX_1:func.3">-SD</a> </span>)</span>  st ( for <font color="Olive" title="b3">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">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_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Olive" title="b1">tx</font>,<font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Olive" title="b2">ty</font>,<font color="Olive" title="b3">i</font> ) holds <br/> <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Olive" title="b1">tx</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Olive" title="b2">ty</font></span><br/><b>let </b><font color="Maroon" title="c3">tx</font>, <font color="Maroon" title="c4">ty</font> be   <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_1.html#K3" title="RADIX_1:func.3">-SD</a> </span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( 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_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c3">tx</font>,<font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c4">ty</font>,<font color="Olive" title="b1">i</font> ) implies  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Maroon" title="c3">tx</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Maroon" title="c4">ty</font> )</span><br/>



<b>assume </b><a NAME="E2:15_2"/><i><font color="Green" title="E16">A7</font></i>: 
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_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c3">tx</font>,<font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c4">ty</font>,<font color="Olive" title="b1">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Maroon" title="c3">tx</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Maroon" title="c4">ty</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">n</font> = <font color="Maroon" title="c1">n</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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<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 <a href="mesfunc1.html#K2" title="MESFUNC1:func.2">K413</a>() =  <a href="radix_1.html#K5" title="RADIX_1:func.5">DigB</a> <font color="Maroon" title="c3">tx</font>,$1;<br/>
<b>consider </b><font color="Maroon" title="c6">txn</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> <b> such that </b><br/><a NAME="E5:15_2"/><i><font color="Green" title="E17">A8</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">txn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font>
 <b>and </b><br/><a NAME="E6:15_2"/><i><font color="Green" title="E18">A9</font></i>: 
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="c6">txn</font> holds <br/><font color="Maroon" title="c6">txn</font> <a href="funct_1.html#K1" title="FUNCT_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>)
 <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="E7:15_2"/><i><font color="Green" title="E19">A10</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">txn</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="c5">n</font>
 
<b>by </b><i><a class="txt" href="#E5:15_2"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:15_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">txn</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">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="15_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">txn</font> or <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a>  )</span><br/>

<b>assume </b><a NAME="E1:15_2_1"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">txn</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a> </span><br/>

<b>then consider </b><font color="Maroon" title="c8">xx</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:15_2_1"/><i><font color="Green" title="E20">A11</font></i>: 
<font color="Maroon" title="c8">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="c6">txn</font>
 <b>and </b><br/><a NAME="E4:15_2_1"/><i><font color="Green" title="E21">A12</font></i>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">txn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">xx</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">xx</font> = <font color="Maroon" title="c8">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>  <b>by </b><i><a class="txt" href="#E3:15_2_1"><i><font color="Green" title="E20">A11</font></i></a></i>;<br/>
<a NAME="E6:15_2_1"/><i><font color="Green" title="E22">A13</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">txn</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="c5">n</font>
 
<b>by </b><i><a class="txt" href="#E5:15_2"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<i><font color="Green" title="E23">A14</font></i>: <font color="Maroon" title="c7">z</font> = 
 <a href="radix_1.html#K5" title="RADIX_1:func.5">DigB</a> <font color="Maroon" title="c3">tx</font>,<font color="Maroon" title="c9">xx</font>
<b>by </b><i><a class="txt" href="#E6:15_2"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="#E3:15_2_1"><i><font color="Green" title="E20">A11</font></i></a>, <a class="txt" href="#E4:15_2_1"><i><font color="Green" title="E21">A12</font></i></a></i>
<br/>.= 
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c3">tx</font>,<font color="Maroon" title="c9">xx</font>
<b>by </b><i><a class="ref" href="radix_1.html#D4" title="RADIX_1:def.4">RADIX_1:def 4</a></i>
;<br/>
<a NAME="E8:15_2_1"/>
<font color="Maroon" title="c9">xx</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="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:15_2_1"><i><font color="Green" title="E20">A11</font></i></a>, <a class="txt" href="#E6:15_2_1"><i><font color="Green" title="E22">A13</font></i></a>, <a class="ref" href="finseq_2.html#T9" title="FINSEQ_2:th.9">FINSEQ_2:9</a></i>;<br/>
<a NAME="E9:15_2_1"/><b>then </b>
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c3">tx</font>,<font color="Maroon" title="c9">xx</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a> 
 
<b>by </b><i><a class="ref" href="radix_1.html#T19" title="RADIX_1:th.19">RADIX_1:19</a></i>;<br/>
<b>hence </b><a NAME="E10:15_2_1"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a> 
 <b>by </b><i><a class="txt" href="#E7:15_2_1"><i><font color="Green" title="E23">A14</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">txn</font> = <font color="Maroon" title="c6">txn</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E10:15_2"/><i><font color="Green" title="E20">A15</font></i>: 
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> <font color="Maroon" title="c5">n</font> holds <br/><font color="Maroon" title="c7">txn</font> <a href="funct_1.html#K1" title="FUNCT_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="c3">tx</font> <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="15_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">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="c8">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="c5">n</font> implies <font color="Maroon" title="c7">txn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">tx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> )</span><br/>

<b>assume </b><a NAME="E1:15_2_2"/><i><font color="Green" title="E21">A16</font></i>: 
<font color="Maroon" title="c8">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="c5">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">txn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">tx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span><br/>

<a NAME="E2:15_2_2"/><b>then </b><i><font color="Green" title="E22">A17</font></i>: 
<font color="Maroon" title="c8">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="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T9" title="FINSEQ_2:th.9">FINSEQ_2:9</a></i>;<br/>
<font color="Maroon" title="c7">txn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> = 
 <a href="radix_1.html#K5" title="RADIX_1:func.5">DigB</a> <font color="Maroon" title="c3">tx</font>,<font color="Maroon" title="c8">i</font>
<b>by </b><i><a class="txt" href="#E6:15_2"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="#E1:15_2_2"><i><font color="Green" title="E21">A16</font></i></a>, <a class="txt" href="#E7:15_2"><i><font color="Green" title="E19">A10</font></i></a></i>
<br/>.= 
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c3">tx</font>,<font color="Maroon" title="c8">i</font>
<b>by </b><i><a class="ref" href="radix_1.html#D4" title="RADIX_1:def.4">RADIX_1:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E4:15_2_2"/>
<font color="Maroon" title="c7">txn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">tx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="#E2:15_2_2"><i><font color="Green" title="E22">A17</font></i></a>, <a class="ref" href="radix_1.html#D3" title="RADIX_1:def.3">RADIX_1:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>2</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 <a href="mesfunc1.html#K2" title="MESFUNC1:func.2">K413</a>() =  <a href="radix_1.html#K5" title="RADIX_1:func.5">DigB</a> <font color="Maroon" title="c4">ty</font>,$1;<br/>
<b>consider </b><font color="Maroon" title="c8">tyn</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> <b> such that </b><br/><a NAME="E12:15_2"/><i><font color="Green" title="E21">A18</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">tyn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font>
 <b>and </b><br/><a NAME="E13:15_2"/><i><font color="Green" title="E22">A19</font></i>: 
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="c8">tyn</font> holds <br/><font color="Maroon" title="c8">tyn</font> <a href="funct_1.html#K1" title="FUNCT_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>)
 <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="E14:15_2"/><i><font color="Green" title="E23">A20</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">tyn</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="c5">n</font>
 
<b>by </b><i><a class="txt" href="#E12:15_2"><i><font color="Green" title="E21">A18</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E15:15_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">tyn</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">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="15_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">tyn</font> or <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a>  )</span><br/>

<b>assume </b><a NAME="E1:15_2_3"/>
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">tyn</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a> </span><br/>

<b>then consider </b><font color="Maroon" title="c10">yy</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:15_2_3"/><i><font color="Green" title="E24">A21</font></i>: 
<font color="Maroon" title="c10">yy</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="c8">tyn</font>
 <b>and </b><br/><a NAME="E4:15_2_3"/><i><font color="Green" title="E25">A22</font></i>: 
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">tyn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">yy</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">yy</font> = <font color="Maroon" title="c10">yy</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>  <b>by </b><i><a class="txt" href="#E3:15_2_3"><i><font color="Green" title="E24">A21</font></i></a></i>;<br/>
<a NAME="E6:15_2_3"/><i><font color="Green" title="E26">A23</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">tyn</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="c5">n</font>
 
<b>by </b><i><a class="txt" href="#E12:15_2"><i><font color="Green" title="E21">A18</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<i><font color="Green" title="E27">A24</font></i>: <font color="Maroon" title="c9">z</font> = 
 <a href="radix_1.html#K5" title="RADIX_1:func.5">DigB</a> <font color="Maroon" title="c4">ty</font>,<font color="Maroon" title="c11">yy</font>
<b>by </b><i><a class="txt" href="#E13:15_2"><i><font color="Green" title="E22">A19</font></i></a>, <a class="txt" href="#E3:15_2_3"><i><font color="Green" title="E24">A21</font></i></a>, <a class="txt" href="#E4:15_2_3"><i><font color="Green" title="E25">A22</font></i></a></i>
<br/>.= 
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c4">ty</font>,<font color="Maroon" title="c11">yy</font>
<b>by </b><i><a class="ref" href="radix_1.html#D4" title="RADIX_1:def.4">RADIX_1:def 4</a></i>
;<br/>
<a NAME="E8:15_2_3"/>
<font color="Maroon" title="c11">yy</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="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:15_2_3"><i><font color="Green" title="E24">A21</font></i></a>, <a class="txt" href="#E6:15_2_3"><i><font color="Green" title="E26">A23</font></i></a>, <a class="ref" href="finseq_2.html#T9" title="FINSEQ_2:th.9">FINSEQ_2:9</a></i>;<br/>
<a NAME="E9:15_2_3"/><b>then </b>
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c4">ty</font>,<font color="Maroon" title="c11">yy</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a> 
 
<b>by </b><i><a class="ref" href="radix_1.html#T19" title="RADIX_1:th.19">RADIX_1:19</a></i>;<br/>
<b>hence </b><a NAME="E10:15_2_3"/>
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a> 
 <b>by </b><i><a class="txt" href="#E7:15_2_3"><i><font color="Green" title="E27">A24</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c9">tyn</font> = <font color="Maroon" title="c8">tyn</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E17:15_2"/><i><font color="Green" title="E24">A25</font></i>: 
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> <font color="Maroon" title="c5">n</font> holds <br/><font color="Maroon" title="c9">tyn</font> <a href="funct_1.html#K1" title="FUNCT_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="c4">ty</font> <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="15_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">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="c10">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="c5">n</font> implies <font color="Maroon" title="c9">tyn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ty</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> )</span><br/>

<b>assume </b><a NAME="E1:15_2_4"/><i><font color="Green" title="E25">A26</font></i>: 
<font color="Maroon" title="c10">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="c5">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">tyn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ty</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span><br/>

<a NAME="E2:15_2_4"/><b>then </b><i><font color="Green" title="E26">A27</font></i>: 
<font color="Maroon" title="c10">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="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T9" title="FINSEQ_2:th.9">FINSEQ_2:9</a></i>;<br/>
<font color="Maroon" title="c9">tyn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> = 
 <a href="radix_1.html#K5" title="RADIX_1:func.5">DigB</a> <font color="Maroon" title="c4">ty</font>,<font color="Maroon" title="c10">i</font>
<b>by </b><i><a class="txt" href="#E13:15_2"><i><font color="Green" title="E22">A19</font></i></a>, <a class="txt" href="#E1:15_2_4"><i><font color="Green" title="E25">A26</font></i></a>, <a class="txt" href="#E14:15_2"><i><font color="Green" title="E23">A20</font></i></a></i>
<br/>.= 
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c4">ty</font>,<font color="Maroon" title="c10">i</font>
<b>by </b><i><a class="ref" href="radix_1.html#D4" title="RADIX_1:def.4">RADIX_1:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E4:15_2_4"/>
<font color="Maroon" title="c9">tyn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ty</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="txt" href="#E2:15_2_4"><i><font color="Green" title="E26">A27</font></i></a>, <a class="ref" href="radix_1.html#D3" title="RADIX_1:def.3">RADIX_1:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c10">txn</font> = <font color="Maroon" title="c7">txn</font> as   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c5">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a> </span>)</span> <b>by </b><i><a class="txt" href="#E5:15_2"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">tyn</font> = <font color="Maroon" title="c9">tyn</font> as   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c5">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a> </span>)</span> <b>by </b><i><a class="txt" href="#E12:15_2"><i><font color="Green" title="E21">A18</font></i></a>, <a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E20:15_2"/><i><font color="Green" title="E25">A28</font></i>: 
<span class="p1">(<span class="default"><a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Maroon" title="c10">txn</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c3">tx</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Maroon" title="c3">tx</font>
 
<b>by </b><i><a class="txt" href="#E10:15_2"><i><font color="Green" title="E20">A15</font></i></a>, <a class="ref" href="radix_2.html#T10" title="RADIX_2:th.10">RADIX_2:10</a></i>;<br/>
<a NAME="E21:15_2"/><i><font color="Green" title="E26">A29</font></i>: 
<span class="p1">(<span class="default"><a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Maroon" title="c11">tyn</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c4">ty</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Maroon" title="c4">ty</font>
 
<b>by </b><i><a class="txt" href="#E17:15_2"><i><font color="Green" title="E24">A25</font></i></a>, <a class="ref" href="radix_2.html#T10" title="RADIX_2:th.10">RADIX_2:10</a></i>;<br/>
<a NAME="E22:15_2"/>
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> <font color="Maroon" title="c5">n</font> holds <br/> <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c10">txn</font>,<font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c11">tyn</font>,<font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">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="c12">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="c5">n</font> implies  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c10">txn</font>,<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c11">tyn</font>,<font color="Maroon" title="c12">i</font> )</span><br/>

<b>assume </b><a NAME="E1:15_2_5"/><i><font color="Green" title="E27">A30</font></i>: 
<font color="Maroon" title="c12">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="c5">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c10">txn</font>,<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c11">tyn</font>,<font color="Maroon" title="c12">i</font></span><br/>

<b>then </b><i><font color="Green" title="E28">A31</font></i>:  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c10">txn</font>,<font color="Maroon" title="c12">i</font> = 
<font color="Maroon" title="c10">txn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
<b>by </b><i><a class="ref" href="radix_1.html#D3" title="RADIX_1:def.3">RADIX_1:def 3</a></i>
<br/>.= 
 <a href="radix_1.html#K5" title="RADIX_1:func.5">DigB</a> <font color="Maroon" title="c3">tx</font>,<font color="Maroon" title="c12">i</font>
<b>by </b><i><a class="txt" href="#E6:15_2"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="#E1:15_2_5"><i><font color="Green" title="E27">A30</font></i></a>, <a class="txt" href="#E7:15_2"><i><font color="Green" title="E19">A10</font></i></a></i>
<br/>.= 
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c3">tx</font>,<font color="Maroon" title="c12">i</font>
<b>by </b><i><a class="ref" href="radix_1.html#D4" title="RADIX_1:def.4">RADIX_1:def 4</a></i>
;<br/>
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c11">tyn</font>,<font color="Maroon" title="c12">i</font> = 
<font color="Maroon" title="c11">tyn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
<b>by </b><i><a class="txt" href="#E1:15_2_5"><i><font color="Green" title="E27">A30</font></i></a>, <a class="ref" href="radix_1.html#D3" title="RADIX_1:def.3">RADIX_1:def 3</a></i>
<br/>.= 
 <a href="radix_1.html#K5" title="RADIX_1:func.5">DigB</a> <font color="Maroon" title="c4">ty</font>,<font color="Maroon" title="c12">i</font>
<b>by </b><i><a class="txt" href="#E13:15_2"><i><font color="Green" title="E22">A19</font></i></a>, <a class="txt" href="#E1:15_2_5"><i><font color="Green" title="E27">A30</font></i></a>, <a class="txt" href="#E14:15_2"><i><font color="Green" title="E23">A20</font></i></a></i>
<br/>.= 
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c4">ty</font>,<font color="Maroon" title="c12">i</font>
<b>by </b><i><a class="ref" href="radix_1.html#D4" title="RADIX_1:def.4">RADIX_1:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E4:15_2_5"/>
 <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c10">txn</font>,<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c11">tyn</font>,<font color="Maroon" title="c12">i</font>
 <b>by </b><i><a class="txt" href="#E2:15_2"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="#E1:15_2_5"><i><font color="Green" title="E27">A30</font></i></a>, <a class="txt" href="#E2:15_2_5"><i><font color="Green" title="E28">A31</font></i></a>, <a class="ref" href="finseq_2.html#T9" title="FINSEQ_2:th.9">FINSEQ_2:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E23:15_2"/><b>then </b><i><font color="Green" title="E27">A32</font></i>: 
 <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Maroon" title="c10">txn</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Maroon" title="c11">tyn</font>
 
<b>by </b><i><a class="txt" href="#E1:15_2"><i><font color="Green" title="E15">A6</font></i></a></i>;<br/>
<a NAME="E24:15_2"/>
<font color="Maroon" title="c5">n</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="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E25:15_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_1.html#K1" title="RADIX_1:func.1">Radix</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c3">tx</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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="c2">k</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Maroon" title="c4">ty</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:15_2"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<b>hence </b><a NAME="E26:15_2"/>
 <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Maroon" title="c3">tx</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Maroon" title="c4">ty</font>
 <b>by </b><i><a class="txt" href="#E20:15_2"><i><font color="Green" title="E25">A28</font></i></a>, <a class="txt" href="#E21:15_2"><i><font color="Green" title="E26">A29</font></i></a>, <a class="txt" href="#E23:15_2"><i><font color="Green" title="E27">A32</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:15"/>
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>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S8" title="NAT_1:sch.8">NAT_1:sch 8</a>(<a class="txt" href="#E1:15"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="#E2:15"><i><font color="Green" title="E14">A5</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:15"/>
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">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b3">tx</font>, <font color="Olive" title="b4">ty</font> being  <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Olive" title="b1">n</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">k</font> <a href="radix_1.html#K3" title="RADIX_1:func.3">-SD</a> </span>)</span>  st ( for <font color="Olive" title="b5">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b5">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="Olive" title="b1">n</font> holds <br/> <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Olive" title="b3">tx</font>,<font color="Olive" title="b5">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K4" title="RADIX_1:func.4">DigA</a> <font color="Olive" title="b4">ty</font>,<font color="Olive" title="b5">i</font> ) holds <br/> <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Olive" title="b3">tx</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="radix_1.html#K8" title="RADIX_1:func.8">SDDec</a> <font color="Olive" title="b4">ty</font>
 ; <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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