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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">a<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">q</font> holds <br/> ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">p</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">q</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> holds <br/><font color="Olive">p</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font> ) );<br/>
<a NAME="E1:16"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="finseq_2.html#M2">Element</a> of 0 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> , <font color="Maroon">q</font> be    <a href="finseq_2.html#M2">Element</a> of 0 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:16_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E2:16_1"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <br/>(  not <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 0 or  not <font color="Maroon">p</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font> or  ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> &amp; not <font color="Maroon">p</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font> ) )
 ;<br/>

<a NAME="E3:16_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K5">&lt;*&gt;</a> <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K5">&lt;*&gt;</a> <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T113">FINSEQ_2:113</a></i>;<br/>
<b>hence </b><a NAME="E4:16_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:16"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:16_2"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">q</font> holds <br/> ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp; <font color="Olive">p</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">q</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> holds <br/><font color="Olive">p</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font> ) )
 ;<br/>

<b>let </b><font color="Maroon">p</font> be    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> , <font color="Maroon">q</font> be    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> ;<br/>

<b>consider </b><font color="Maroon">t1</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> , <font color="Maroon">d1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:16_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font> <a href="polynom3.html#K3" target="_self">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T137">FINSEQ_2:137</a></i>;<br/>
<b>consider </b><font color="Maroon">t2</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> , <font color="Maroon">d2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:16_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> <a href="polynom3.html#K3" target="_self">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T137">FINSEQ_2:137</a></i>;<br/>
<b>assume </b><a NAME="E6:16_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E7:16_2"/><i><font color="Green">E64</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">t2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_2_1"/>
( <font color="Maroon">t1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">t2</font> or <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1"><b>suppose </b></a><a NAME="E1:16_2_1_1"/>
<font color="Maroon">t1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">t2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:16_2_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E4:16_2_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">t1</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">t2</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E5:16_2_1_1"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> holds <br/><font color="Maroon">t1</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font>
 <b>by </b><i/>;<br/><b>take </b>
<font color="Maroon">i</font>
;<br/><b>thus </b><a NAME="E6:16_2_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T9">FINSEQ_2:9</a></i>;<br/><a NAME="E7:16_2_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">t1</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">t2</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E8:16_2_1_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t1</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">q</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E9:16_2_1_1"/>
<font color="Maroon">p</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E10:16_2_1_1"/><i><font color="Green">E69</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E11:16_2_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 ;<br/><a NAME="E12:16_2_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">t1</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E13:16_2_1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E14:16_2_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E15:16_2_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">t1</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">t2</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><font color="Maroon">p</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">t2</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font>
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font>
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_2"><b>suppose </b></a><a NAME="E1:16_2_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">i</font> = <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1;<br/><b>thus </b><a NAME="E2:16_2_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/><a NAME="E3:16_2_1_2"/>
( <font color="Maroon">p</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d1</font> &amp; <font color="Maroon">q</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d2</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_2.html#T136">FINSEQ_2:136</a></i>;<br/><b>hence </b><a NAME="E4:16_2_1_2"/>
<font color="Maroon">p</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , , </i>;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E5:16_2_1_2"/><i><font color="Green">E74</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E6:16_2_1_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 ;<br/><a NAME="E7:16_2_1_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:16_2_1_2"/><b>then </b><i><font color="Green">E81</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">t1</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">t2</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><font color="Maroon">p</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">t2</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font>
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font>
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>thus </b><a NAME="E3:16"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


</div>
