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<b>let </b><font color="Maroon" title="c1">i</font> be    <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> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <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>means </b><a href="comput_1.html#K17" title="COMPUT_1:func.17">[*]</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">i</font>,$1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> $1;<br/>
<a NAME="E1:161"/><i><font color="Green" title="E104">A1</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="161_1"><b>proof </b></a><div class="add">

<b>thus </b><a href="comput_1.html#K17" title="COMPUT_1:func.17">[*]</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">i</font>,0</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> = 
<span class="p1">(<span class="default">1 <a href="comput_1.html#K5" title="COMPUT_1:func.5">const</a> 0</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">i</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
<b>by </b><i><a class="ref" href="comput_1.html#T83" target="_self" title="COMPUT_1:th.83">Th83</a></i>
<br/>.= 
<font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 0
<b>by </b><i><a class="txt" href="comput_1.html#E46"><i><font color="Green" title="E36">Lm2</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E105">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="161_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">j</font> be    <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> ;<br/><b>assume </b><a NAME="E1:161_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">j</font>]
 ;<br/><b>then </b><a href="comput_1.html#K17" title="COMPUT_1:func.17">[*]</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> = 
<span class="p1">(<span class="default"><a href="comput_1.html#K15" title="COMPUT_1:func.15">(1,2)-&gt;(1,?,2)</a> <a href="comput_1.html#K16" title="COMPUT_1:func.16">[+]</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">j</font></span>)</span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
<b>by </b><i><a class="ref" href="comput_1.html#T85" target="_self" title="COMPUT_1:th.85">Th85</a></i>
<br/>.= 
<a href="comput_1.html#K16" title="COMPUT_1:func.16">[+]</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">j</font></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
<b>by </b><i><a class="ref" href="comput_1.html#T87" target="_self" title="COMPUT_1:th.87">Th87</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">j</font></span>)</span>
<b>by </b><i><a class="ref" href="comput_1.html#T89" target="_self" title="COMPUT_1:th.89">Th89</a></i>
<br/>.= 
<font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>

;<br/><b>hence </b><a NAME="E3:161_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E3:161"/>
for <font color="Olive" title="b1">i</font> being   <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>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="comput_1.html#E1:161"><i><font color="Green" title="E104">A1</font></i></a>, <a class="txt" href="comput_1.html#E2:161"><i><font color="Green" title="E105">A2</font></i></a>);</i><br/>


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