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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> <a href="binarith.html#K7">Absval</a> <span class="p1">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1;<br/>
<a NAME="E1:11"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<b>by </b><i>, <a class="ref" href="binarith.html#T42">BINARITH:42</a></i>;<br/>
<a NAME="E2:11"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">n</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:11_1"/><i><font color="Green">E33</font></i>: 
 <a href="binarith.html#K7">Absval</a> <span class="p1">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<b>thus </b> <a href="binarith.html#K7">Absval</a> <span class="p1">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
 <a href="binarith.html#K7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K10">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="binari_2.html#T9">BINARI_2:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K7">Absval</a> <span class="p2">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">n</font></span>)</span></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">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T46">BINARITH:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K7">Absval</a> <span class="p2">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 0
<b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
<br/>.= 
1
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:11"/>
for <font color="Olive">n</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="binarith.html#S1">BINARITH:sch 1</a>(, );</i><br/>


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