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<div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:176"/><i><font color="Green">E48</font></i>: 
 <a href="comput_1.html#K3" target="_self">arity</a> <span class="p1">(<span class="default">0 <a href="comput_1.html#K6" target="_self">const</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E2:176"/>
0 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:176"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> 0 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>thus </b><i><font color="Green">E51</font></i>: <a href="comput_1.html#K21" target="_self">[pred]</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span> = 
<span class="p1">(<span class="default">0 <a href="comput_1.html#K6" target="_self">const</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, </i>
<br/>.= 
0
<b>by </b><i>, </i>
;<br/>

<a NAME="E5:176"/><i><font color="Green">E52</font></i>: 
 <a href="comput_1.html#K3" target="_self">arity</a> <span class="p1">(<span class="default">2 <a href="comput_1.html#K8" target="_self">proj</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">p0</font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">0,0</span><a href="finseq_4.html#K2">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of 2 <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#T121">FINSEQ_2:121</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><a href="comput_1.html#K21" target="_self">[pred]</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E7:176"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="176_2"><b>proof </b></a><div class="add">

<b>thus </b><a href="comput_1.html#K21" target="_self">[pred]</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> = 
<span class="p1">(<span class="default">2 <a href="comput_1.html#K8" target="_self">proj</a> 1</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p0</font>
<b>by </b><i>, <a class="ref" href="comput_1.html#T25" target="_self">Th25</a>, <a class="ref" href="comput_1.html#T26" target="_self">Th26</a>, </i>
<br/>.= 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">0,0</span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i/>
<br/>.= 
0
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="176_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:176_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">i</font>]
 ;<br/><b>reconsider </b><font color="Maroon">p0</font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">i</font></span><a href="finseq_4.html#K2">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of 2 <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#T121">FINSEQ_2:121</a></i>;<br/><a href="comput_1.html#K21" target="_self">[pred]</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> = 
<span class="p1">(<span class="default">2 <a href="comput_1.html#K8" target="_self">proj</a> 1</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p0</font>
<b>by </b><i>, <a class="ref" href="comput_1.html#T26" target="_self">Th26</a>, <a class="ref" href="comput_1.html#D7" target="_self">Def7</a>, </i>
<br/>.= 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">i</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i/>
<br/>.= 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><b>hence </b><a NAME="E4:176_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E9:176"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="comput_1.html#T28" target="_self">Th28</a>, <a class="ref" href="comput_1.html#D6" target="_self">Def6</a>);<br/></i>
<b>hence </b><a NAME="E10:176"/>
<a href="comput_1.html#K21" target="_self">[pred]</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 ;<br/>


</div>
