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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b> <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1;<br/>
<a NAME="E1:11"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<b>by </b><i><a class="ref" href="binari_3.html#T11" target="_self" title="BINARI_3:th.11">Th11</a>, <a class="ref" href="binarith.html#T42" title="BINARITH:th.42">BINARITH:42</a></i>;<br/>
<a NAME="E2:11"/><i><font color="Green" title="E9">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <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="11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">n</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:11_1"/><i><font color="Green" title="E10">A3</font></i>: 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>

<b>thus </b> <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <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></span>)</span> = 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="binari_2.html#T9" title="BINARI_2:th.9">BINARI_2:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p2">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2" title="CQC_LANG:func.2">IFEQ</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T46" title="BINARITH:th.46">BINARITH:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p2">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0
<b>by </b><i><a class="ref" href="cqc_lang.html#D1" title="CQC_LANG:def.1">CQC_LANG:def 1</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="binari_3.html#E1:11_1"><i><font color="Green" title="E10">A3</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:11"/>
for <font color="Olive" title="b1">n</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="binarith.html#S1" title="BINARITH:sch.1">BINARITH:sch 1</a>(<a class="txt" href="binari_3.html#E1:11"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="binari_3.html#E2:11"><i><font color="Green" title="E9">A2</font></i></a>);</i><br/>


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