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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">p</font> being <a href="yellow_1.html#V1">RelStr-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">a<sub>1</sub></font>  st ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">a<sub>1</sub></font> holds <br/> ( <font color="Olive">p</font> <a href="yellow_1.html#K4">.</a> <font color="Olive">i</font> is <a href="dickson.html#V4" target="_self">Dickson</a> &amp; <font color="Olive">p</font> <a href="yellow_1.html#K4">.</a> <font color="Olive">i</font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a> ) ) holds <br/>(  <a href="yellow_1.html#K5">product</a> <font color="Olive">p</font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a> &amp;  <a href="yellow_1.html#K5">product</a> <font color="Olive">p</font> is <a href="dickson.html#V4" target="_self">Dickson</a> );<br/>
<a NAME="E1:62"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be  <a href="yellow_1.html#V1">RelStr-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of 1;<br/>

<b>assume </b><a NAME="E1:62_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of 1 holds <br/> ( <font color="Maroon">p</font> <a href="yellow_1.html#K4">.</a> <font color="Olive">i</font> is <a href="dickson.html#V4" target="_self">Dickson</a> &amp; <font color="Maroon">p</font> <a href="yellow_1.html#K4">.</a> <font color="Olive">i</font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a> )
 ;<br/>

<b>reconsider </b><font color="Maroon">z</font> = 0 as    <a href="subset_1.html#M1">Element</a> of 1 <b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:62_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">p</font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">z</font> is <a href="dickson.html#V4" target="_self">Dickson</a>
 
<b>by </b><i/>;<br/>
<a NAME="E4:62_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">p</font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">z</font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a>
 
<b>by </b><i/>;<br/>
<a NAME="E5:62_1"/>
<font color="Maroon">p</font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">z</font>, <a href="yellow_1.html#K5">product</a> <font color="Maroon">p</font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:62_1"/>
(  <a href="yellow_1.html#K5">product</a> <font color="Maroon">p</font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a> &amp;  <a href="yellow_1.html#K5">product</a> <font color="Maroon">p</font> is <a href="dickson.html#V4" target="_self">Dickson</a> )
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:62_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/><b>thus </b><a NAME="E2:62_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be  <a href="yellow_1.html#V1">RelStr-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1;<br/>

<b>assume </b><a NAME="E1:62_2_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 holds <br/> ( <font color="Maroon">p</font> <a href="yellow_1.html#K4">.</a> <font color="Olive">i</font> is <a href="dickson.html#V4" target="_self">Dickson</a> &amp; <font color="Maroon">p</font> <a href="yellow_1.html#K4">.</a> <font color="Olive">i</font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a> )
 ;<br/>

<a NAME="E2:62_2_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">ns</font> = <font color="Maroon">n</font> as   <a href="subset_1.html#NM2">Subset</a> of <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="card_1.html#T56">CARD_1:56</a></i>;<br/>
<a NAME="E4:62_2_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 </span>}</span> 
 
<b>by </b><i><a class="ref" href="axioms.html#T30">AXIOMS:30</a></i>;<br/>
<a NAME="E5:62_2_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:62_2_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="dickson.html#D2" target="_self">Def2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">ne</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 ;<br/>
<b>reconsider </b><font color="Maroon">pns</font> = <font color="Maroon">p</font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">ns</font> as  <a href="yellow_1.html#V1">RelStr-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">n</font> ;<br/>
<a NAME="E9:62_2_1"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n</font> holds <br/> ( <font color="Maroon">pns</font> <a href="yellow_1.html#K4">.</a> <font color="Olive">i</font> is <a href="dickson.html#V4" target="_self">Dickson</a> &amp; <font color="Maroon">pns</font> <a href="yellow_1.html#K4">.</a> <font color="Olive">i</font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n</font>;<br/>

<a NAME="E1:62_2_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">pns</font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E2:62_2_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="axioms.html#T30">AXIOMS:30</a></i>;<br/>
<a NAME="E3:62_2_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font>
 
;<br/>
<b>then consider </b><font color="Maroon">k</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="E5:62_2_1_1"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/>
<a NAME="E6:62_2_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E7:62_2_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="dickson.html#D2" target="_self">Def2</a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 ;<br/>
<a NAME="E9:62_2_1_1"/>
<font color="Maroon">i'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 
;<br/>
<b>hence </b><a NAME="E10:62_2_1_1"/>
( <font color="Maroon">pns</font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">i</font> is <a href="dickson.html#V4" target="_self">Dickson</a> &amp; <font color="Maroon">pns</font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">i</font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a> )
 <b>by </b><i>, <a class="ref" href="dickson.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:62_2_1"/><b>then </b><i><font color="Green">E71</font></i>: 
(  <a href="yellow_1.html#K5">product</a> <font color="Maroon">pns</font> is <a href="dickson.html#V4" target="_self">Dickson</a> &amp;  <a href="yellow_1.html#K5">product</a> <font color="Maroon">pns</font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E11:62_2_1"/>
( <font color="Maroon">p</font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">ne</font> is <a href="dickson.html#V4" target="_self">Dickson</a> &amp; <font color="Maroon">p</font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">ne</font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E12:62_2_1"/><b>then </b><i><font color="Green">E72</font></i>: 
( <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">ns</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">ne</font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span> is <a href="dickson.html#V4" target="_self">Dickson</a> &amp; <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">ns</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">ne</font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span> is <a href="dickson.html#V3" target="_self">quasi_ordered</a> )
 
<b>by </b><i><a class="ref" href="dickson.html#T4" target="_self">Th4</a>, </i>;<br/>
<a NAME="E13:62_2_1"/>
<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">ns</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">ne</font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span>, <a href="yellow_1.html#K5">product</a> <font color="Maroon">p</font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E14:62_2_1"/>
(  <a href="yellow_1.html#K5">product</a> <font color="Maroon">p</font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a> &amp;  <a href="yellow_1.html#K5">product</a> <font color="Maroon">p</font> is <a href="dickson.html#V4" target="_self">Dickson</a> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>thus </b><a NAME="E3:62"/>
for <font color="Olive">n</font> being non <a href="xboole_0.html#V1">empty</a>  <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="binarith.html#S1">BINARITH:sch 1</a>(<a class="ref" href="dickson.html#T3" target="_self">Th3</a>, );</i><br/>


</div>
