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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>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <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> as    <a href="finseq_2.html#M2">Element</a> of 1 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">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#K17" 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="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E2:161"/><i><font color="Green">E105</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" 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#K8" target="_self">proj</a> 1</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="comput_1.html#T83" target="_self">Th83</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="comput_1.html#K2" target="_self">.</a> 1
<b>by </b><i><a class="ref" href="comput_1.html#T42" target="_self">Th42</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 0
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E106</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">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <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>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of 3 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T124">FINSEQ_2:124</a></i>;<br/><b>assume </b><a NAME="E2:161_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 ;<br/><b>then </b><a href="comput_1.html#K17" 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>4</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">3 <a href="comput_1.html#K7" target="_self">succ</a> 3</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="comput_1.html#T85" target="_self">Th85</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="comput_1.html#D10" target="_self">Def10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>

;<br/><b>hence </b><a NAME="E4:161_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E4:161"/>
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#E2:161"><i><font color="Green">E105</font></i></a>, <a class="txt" href="comput_1.html#E3:161"><i><font color="Green">E106</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:161"/>
<a href="comput_1.html#K17" 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="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


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