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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being  <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">a<sub>1</sub></font>,<a href="margrel1.html#K6">BOOLEAN</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="binarith.html#K9">Absval</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0;<br/>
<a NAME="E1:6"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="binarith.html#NM1">Tuple</a> of 1,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/>

<b>assume </b><a NAME="E1:6_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> 1
 ;<br/>

<b>then </b><font color="Maroon">c<sub>1</sub></font> = 
1 <a href="finseqop.html#K4">|-&gt;</a> 0

<br/>.= 
<span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="binarith.html#K13">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T73">FINSEQ_2:73</a>, <a class="ref" href="margrel1.html#T36">MARGREL1:36</a></i>
;<br/>
<b>hence </b><a NAME="E3:6_1"/>
 <a href="binarith.html#K9">Absval</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="binarith.html#T41">BINARITH:41</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:6"/><i><font color="Green">E8</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:6_2"/><i><font color="Green">E9</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="margrel1.html#K6">BOOLEAN</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font> holds <br/> <a href="binarith.html#K9">Absval</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="binarith.html#NM1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/>

<a NAME="E2:6_2"/>
 <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <a href="margrel1.html#K6">BOOLEAN</a>  <a href="finseq_1.html#K13">*</a> 
 
<b>by </b><i><a class="ref" href="binari_3.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E3:6_2"/><b>then </b><i><font color="Green">E10</font></i>: 
 <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="margrel1.html#K6">BOOLEAN</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E4:6_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font> as   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="margrel1.html#K6">BOOLEAN</a>  <b>by </b><i><a class="txt" href="binari_3.html#E3:6_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<b>assume </b><a NAME="E6:6_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<b>hence </b> <a href="binarith.html#K9">Absval</a> <font color="Maroon">c<sub>2</sub></font> = 
 <a href="binarith.html#K9">Absval</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K12">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="binarith.html#K13">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="binari_3.html#T4" target="_self">Th4</a>, <a class="ref" href="margrel1.html#T36">MARGREL1:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K9">Absval</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <a href="margrel1.html#K7">FALSE</a> ,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T46">BINARITH:46</a></i>
<br/>.= 
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <a href="margrel1.html#K7">FALSE</a> ,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binari_3.html#E1:6_2"><i><font color="Green">E9</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:6"/>
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="binarith.html#S1">BINARITH:sch 1</a>(<a class="txt" href="binari_3.html#E1:6"><i><font color="Green">E7</font></i></a>, <a class="txt" href="binari_3.html#E2:6"><i><font color="Green">E8</font></i></a>);</i><br/>


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