<?xml version="1.0"?>
<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  <a href="xboole_0.html#K1">{}</a> ;<br/>

<a NAME="E1:63"/><i><font color="Green">E46</font></i>: 
 <a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</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>,<span class="p1">(<span class="default"><a href="cqc_sim1.html#K13">id</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="yellow_1.html#T26">YELLOW_1:26</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E2:63"/><i><font color="Green">E47</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K13">id</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><a class="ref" href="yellow_1.html#T26">YELLOW_1:26</a></i>;<br/>
<b>thus </b><a NAME="E3:63"/>
not  <a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i><a class="ref" href="yellow_1.html#T26">YELLOW_1:26</a></i>;<br/>

<b>thus </b><a NAME="E4:63"/>
 <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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:63_1"/>
 <a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>1</sub></font> is <a href="orders_2.html#V2">reflexive</a>
 <b>by </b><i><a class="ref" href="yellow_1.html#T26">YELLOW_1:26</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="dickson.html#D3" target="_self">DICKSON:def 3</a><br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a>,<a class="ref" href="orders_2.html#D5">ORDERS_2:def 5</a><br/>

<b>assume </b><a NAME="E2:63_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 ;<br/>

<b>hence </b><a NAME="E3:63_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E2:63"><i><font color="Green">E47</font></i></a>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:63"/>
 <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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="dickson.html#D10" target="_self">DICKSON:def 10</a><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_2_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>5</sub></font> <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> )
 </span><b>by </b><i><a class="txt" href="dickson.html#E1:63"><i><font color="Green">E46</font></i></a>, <a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_1_1"><b>suppose </b></a><a NAME="E1:63_2_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>6</sub></font> =  <a href="xboole_0.html#K1">{}</a> ;<br/><a NAME="E2:63_2_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E1:63_2_1_1"><i><font color="Green">E48</font></i></a></i>;<br/><b>hence </b><a NAME="E3:63_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">c<sub>5</sub></font>, <a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="dickson.html#T26" target="_self">Th26</a></i>;<br/><b>thus </b><a NAME="E4:63_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="finset_1.html#V1">finite</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_1_2"><b>suppose </b></a><a NAME="E1:63_2_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <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>
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>6</sub></font> = <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>;<br/><b>thus </b><a NAME="E2:63_2_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">c<sub>5</sub></font>, <a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:63_2_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:63_2_1_2"><i><font color="Green">E48</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="dickson.html#D9" target="_self">DICKSON:def 9</a><br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E2:63_2_1_2_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font>;<br/>

<b>thus </b><a NAME="E3:63_2_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:63_2_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="dickson.html#E2:63_2_1_2_1"><i><font color="Green">E49</font></i></a></i>;<br/>

<a NAME="E4:63_2_1_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K13">id</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><a class="txt" href="dickson.html#E1:63_2_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="dickson.html#E2:63_2_1_2_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E5:63_2_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E2:63"><i><font color="Green">E47</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:63_2_1_2"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="finset_1.html#V1">finite</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>thus </b><a NAME="E6:63"/>
 <a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>1</sub></font> is <a href="orders_2.html#V4">antisymmetric</a>
 <b>by </b><i><a class="ref" href="yellow_1.html#T26">YELLOW_1:26</a></i>;<br/>


</div>
