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

<b>let </b><font color="Maroon">L</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_0.html#V24">up-complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>



<b>thus </b><a NAME="E1:31"/>
( <font color="Maroon">X</font> is <a href="waybel_6.html#V4" target="_self">order-generating</a> implies for <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>  st <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; ( for <font color="Olive">Z</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Olive">Y</font> holds   <a href="yellow_0.html#K2">"/\"</a> <font color="Olive">Z</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) holds <br/>the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Olive">Y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">X</font> is <a href="waybel_6.html#V4" target="_self">order-generating</a>
 ;<br/>

<b>let </b><font color="Maroon">Y</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:31_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E3:31_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">Z</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> holds   <a href="yellow_0.html#K2">"/\"</a> <font color="Olive">Z</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">l1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:31_1_1"/>
<font color="Maroon">l1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 ;<br/><b>then reconsider </b><font color="Maroon">l</font> = <font color="Maroon">l1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/><a NAME="E3:31_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:31_1_1"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E5:31_1_1"/>
<span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E6:31_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E7:31_1_1"/>
<font color="Maroon">l1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:31_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E6:31_1"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 ;<br/>


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

<b>thus </b><a NAME="E2:31"/>
( ( for <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>  st <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; ( for <font color="Olive">Z</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Olive">Y</font> holds   <a href="yellow_0.html#K2">"/\"</a> <font color="Olive">Z</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) holds <br/>the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Olive">Y</font> ) implies <font color="Maroon">X</font> is <a href="waybel_6.html#V4" target="_self">order-generating</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_2"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>  st <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; ( for <font color="Olive">Z</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Olive">Y</font> holds   <a href="yellow_0.html#K2">"/\"</a> <font color="Olive">Z</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) holds <br/>the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Olive">Y</font>
 ;<br/>

<b>set </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Olive">Z</font>,<font color="Maroon">L</font></span>)</span> where <font color="Olive">Z</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Olive">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:31_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Olive">Z</font>,<font color="Maroon">L</font></span>)</span> where <font color="Olive">Z</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Olive">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">Z</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:31_2_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Z</font>,<font color="Maroon">L</font> &amp; <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 ;<br/><b>thus </b><a NAME="E4:31_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Olive">Z</font>,<font color="Maroon">L</font></span>)</span> where <font color="Olive">Z</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Olive">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:31_2_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/><a NAME="E3:31_2_2"/><i><font color="Green">E79</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x1</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>reconsider </b><font color="Maroon">x2</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x1</font></span><a href="domain_1.html#K6">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/><a NAME="E5:31_2_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">x2</font>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T39">YELLOW_0:39</a></i>;<br/><b>hence </b><a NAME="E6:31_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E5:31_2"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E6:31_2"/>
for <font color="Olive">l</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  ex <font color="Olive">Z</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st <font color="Olive">l</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Olive">Z</font>,<font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">l</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<a NAME="E1:31_2_3"/>
for <font color="Olive">Z</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> holds   <a href="yellow_0.html#K2">"/\"</a> <font color="Olive">Z</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Z</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> );<br/>
<b>set </b><font color="Maroon">N</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Olive">A</font>,<font color="Maroon">L</font></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>] </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:31_2_3_1_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/><a NAME="E2:31_2_3_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><b>then consider </b><font color="Maroon">Z</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:31_2_3_1_1"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Z</font>,<font color="Maroon">L</font> &amp; <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 ;<br/><b>thus </b><a NAME="E5:31_2_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Olive">A</font>,<font color="Maroon">L</font></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>] </span>}</span> 
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E2:31_2_3_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Olive">A</font>,<font color="Maroon">L</font></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>] </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:31_2_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Olive">A</font>,<font color="Maroon">L</font></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>] </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">S</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:31_2_3_1_2"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">S</font>,<font color="Maroon">L</font> &amp; <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">S</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 ;<br/><b>thus </b><a NAME="E4:31_2_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E4:31_2_3_1"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Olive">A</font>,<font color="Maroon">L</font></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>] </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then </b><i><font color="Green">E88</font></i>:  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Z</font>,<font color="Maroon">L</font> = 
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Olive">A</font>,<font color="Maroon">L</font></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>] </span>}</span> ,<font color="Maroon">L</font>
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
<br/>.= 
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>] </span>}</span> </span>)</span>,<font color="Maroon">L</font>
<b>from </b><i><a class="ref" href="yellow_3.html#S4">YELLOW_3:sch 4</a>()</i>
;<br/>
<b>set </b><font color="Maroon">S</font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Olive">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="yellow_0.html#K2">"/\"</a> <font color="Olive">A</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> ) </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:31_2_3_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Olive">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="yellow_0.html#K2">"/\"</a> <font color="Olive">A</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:31_2_3_1_4"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E4:31_2_3_1_4"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Olive">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="yellow_0.html#K2">"/\"</a> <font color="Olive">A</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:31_2_3_1_4"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">A</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E7:31_2_3_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">S</font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Olive">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="yellow_0.html#K2">"/\"</a> <font color="Olive">A</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> ) </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:31_2_3_1_5"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Olive">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="yellow_0.html#K2">"/\"</a> <font color="Olive">A</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:31_2_3_1_5"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">A</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 ;<br/><b>thus </b><a NAME="E4:31_2_3_1_5"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E9:31_2_3_1"/><b>then </b>
<font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T94">ZFMISC_1:94</a></i>;<br/>
<b>hence </b><a NAME="E10:31_2_3_1"/>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Z</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:31_2_3"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:31_2_3"/><b>then </b>
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
;<br/>
<b>then consider </b><font color="Maroon">Z</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:31_2_3"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Z</font>,<font color="Maroon">L</font> &amp; <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 ;<br/>
<b>thus </b><a NAME="E6:31_2_3"/>
 ex <font color="Olive">Z</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st <font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Olive">Z</font>,<font color="Maroon">L</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:31_2"/>
<font color="Maroon">X</font> is <a href="waybel_6.html#V4" target="_self">order-generating</a>
 <b>by </b><i/>;<br/>


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


</div>
