<?xml version="1.0"?>
<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></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">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">a<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Olive">b<sub>2</sub></font> is <a href="dickson.html#V4" target="_self">Dickson</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a> &amp;  <a href="yellow_1.html#K5">product</a> <font color="Olive">b<sub>1</sub></font> is <a href="dickson.html#V4" target="_self">Dickson</a> );<br/>
<a NAME="E1:62"/><i><font color="Green">E45</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">c<sub>1</sub></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">E46</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of 1 holds <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="dickson.html#V4" target="_self">Dickson</a> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a> )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>2</sub></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">E47</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>2</sub></font> is <a href="dickson.html#V4" target="_self">Dickson</a>
 
<b>by </b><i><a class="txt" href="dickson.html#E1:62_1"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E4:62_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>2</sub></font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a>
 
<b>by </b><i><a class="txt" href="dickson.html#E1:62_1"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E5:62_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>2</sub></font>, <a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>1</sub></font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="dickson.html#T40" target="_self">Th40</a></i>;<br/>
<b>hence </b><a NAME="E6:62_1"/>
(  <a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>1</sub></font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a> &amp;  <a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>1</sub></font> is <a href="dickson.html#V4" target="_self">Dickson</a> )
 <b>by </b><i><a class="txt" href="dickson.html#E3:62_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="dickson.html#E4:62_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="dickson.html#T39" target="_self">Th39</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E46</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">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:62_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>]
 ;<br/><b>thus </b><a NAME="E2:62_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be  <a href="yellow_1.html#V1">RelStr-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1;<br/>

<b>assume </b><a NAME="E1:62_2_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 holds <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="dickson.html#V4" target="_self">Dickson</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a> )
 ;<br/>

<a NAME="E2:62_2_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></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">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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">E49</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="dickson.html#E4:62_2_1"><i><font color="Green">E49</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font> as  <a href="yellow_1.html#V1">RelStr-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E9:62_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>5</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="dickson.html#V4" target="_self">Dickson</a> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:62_2_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></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">E51</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></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">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:62_2_1_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="dickson.html#E2:62_2_1_1"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E6:62_2_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="dickson.html#E5:62_2_1_1"><i><font color="Green">E52</font></i></a>, <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">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="dickson.html#E4:62_2_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="dickson.html#E5:62_2_1_1"><i><font color="Green">E52</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 ;<br/>
<a NAME="E9:62_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
;<br/>
<b>hence </b><a NAME="E10:62_2_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>6</sub></font> is <a href="dickson.html#V4" target="_self">Dickson</a> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>6</sub></font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a> )
 <b>by </b><i><a class="txt" href="dickson.html#E1:62_2_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="dickson.html#E1:62_2_1_1"><i><font color="Green">E50</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:62_2_1"/><b>then </b><i><font color="Green">E50</font></i>: 
(  <a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>5</sub></font> is <a href="dickson.html#V4" target="_self">Dickson</a> &amp;  <a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>5</sub></font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a> )
 
<b>by </b><i><a class="txt" href="dickson.html#E1:62_2"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E11:62_2_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font> is <a href="dickson.html#V4" target="_self">Dickson</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a> )
 
<b>by </b><i><a class="txt" href="dickson.html#E1:62_2_1"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E12:62_2_1"/><b>then </b><i><font color="Green">E51</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">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></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="txt" href="dickson.html#E10:62_2_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="dickson.html#T38" target="_self">Th38</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">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span>, <a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="dickson.html#T42" target="_self">Th42</a></i>;<br/>
<b>hence </b><a NAME="E14:62_2_1"/>
(  <a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a> &amp;  <a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font> is <a href="dickson.html#V4" target="_self">Dickson</a> )
 <b>by </b><i><a class="txt" href="dickson.html#E12:62_2_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="dickson.html#T39" target="_self">Th39</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>thus </b><a NAME="E3:62"/>
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="binarith.html#S1">BINARITH:sch 1</a>(<a class="txt" href="dickson.html#E1:62"><i><font color="Green">E45</font></i></a>, <a class="txt" href="dickson.html#E2:62"><i><font color="Green">E46</font></i></a>);</i><br/>


</div>
