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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <a href="comput_1.html#K20" target="_self">[^]</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">a<sub>1</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="card_4.html#K1">|^</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:174"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="174_1"><b>proof </b></a><div class="add">

<b>thus </b><a href="comput_1.html#K20" target="_self">[^]</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,0</span><a href="finseq_4.html#K2">*&gt;</a></span> = 
<span class="p1">(<span class="default">1 <a href="comput_1.html#K6" target="_self">const</a> 1</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>
<b>by </b><i><a class="ref" href="comput_1.html#T83" target="_self">Th83</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="comput_1.html#T36" target="_self">Th36</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="card_4.html#K1">|^</a> 0
<b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>
;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E107</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="174_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><b>assume </b><a NAME="E2:174_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 ;<br/><b>then </b><a href="comput_1.html#K20" target="_self">[^]</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> = 
<span class="p1">(<span class="default"><a href="comput_1.html#K16" target="_self">(1,2)-&gt;(1,?,2)</a> <a href="comput_1.html#K18" target="_self">[*]</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="finseq_4.html#K3">*&gt;</a></span>
<b>by </b><i><a class="ref" href="comput_1.html#T85" target="_self">Th85</a></i>
<br/>.= 
<a href="comput_1.html#K18" target="_self">[*]</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span>
<b>by </b><i><a class="ref" href="comput_1.html#T87" target="_self">Th87</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="comput_1.html#T90" target="_self">Th90</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="card_4.html#K1">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
;<br/><b>hence </b><a NAME="E4:174_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E3:174"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="comput_1.html#E1:174"><i><font color="Green">E106</font></i></a>, <a class="txt" href="comput_1.html#E2:174"><i><font color="Green">E107</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:174"/>
<a href="comput_1.html#K20" target="_self">[^]</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


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