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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means  <a href="binarith.html#K9">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">E9</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">Th11</a>, <a class="ref" href="binarith.html#T42">BINARITH:42</a></i>;<br/>
<a NAME="E2:11"/><i><font color="Green">E10</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="11_1"><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="E1:11_1"/><i><font color="Green">E11</font></i>: 
 <a href="binarith.html#K9">Absval</a> <span class="p1">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<b>thus </b> <a href="binarith.html#K9">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">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
 <a href="binarith.html#K9">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">c<sub>1</sub></font></span>)</span> <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_2.html#T9">BINARI_2:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K9">Absval</a> <span class="p2">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></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">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/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K9">Absval</a> <span class="p2">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></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><a class="txt" href="binari_3.html#E1:11_1"><i><font color="Green">E11</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:11"/>
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#E1:11"><i><font color="Green">E9</font></i></a>, <a class="txt" href="binari_3.html#E2:11"><i><font color="Green">E10</font></i></a>);</i><br/>


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