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

<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="struct_0.html#L1">1-sorted</a> ;<br/><b>let </b><font color="Maroon">N</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">F</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:11"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">A</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F</font> holds  <font color="Olive">A</font> is    <a href="yellow19.html#M1" target="_self">Subset</a> of <font color="Maroon">S</font>,<font color="Maroon">N</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> st <br/>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K6">|</a> <font color="Olive">i</font></span>)</span> );<br/>
<div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_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="E1:11_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">x</font> as    <a href="yellow19.html#M1" target="_self">Subset</a> of <font color="Maroon">S</font>,<font color="Maroon">N</font> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">i</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E4:11_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">i</font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">y</font>;<br/><b>thus </b><a NAME="E6:11_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>
 ;<br/><b>thus </b><a NAME="E7:11_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:11"/><i><font color="Green">E40</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> )
 <b>and </b><br/><a NAME="E5:11"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">B</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">N</font> <b>by </b><i>, <a class="ref" href="finset_1.html#T26">FINSET_1:26</a></i>;<br/>
<a NAME="E7:11"/>
(  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">N</font> is <a href="waybel_0.html#V1">directed</a> &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a>, <a class="ref" href="waybel_0.html#D6">WAYBEL_0:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">j</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E9:11"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E10:11"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">j</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T1">WAYBEL_0:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">C</font> =  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j</font></span>)</span> as    <a href="yellow19.html#M1" target="_self">Subset</a> of <font color="Maroon">S</font>,<font color="Maroon">N</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">C</font>
;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E12:11"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<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="E14:11"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E15:11"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E17:11"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j</font></span>)</span> <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="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">j</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">k</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel_9.html#T12">WAYBEL_9:12</a></i>;<br/>
<b>then consider </b><font color="Maroon">k</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E19:11"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">j</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_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="E1:11_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><b>then consider </b><font color="Maroon">i</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E3:11_2"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">i</font></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E4:11_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:11_2"/><b>then </b>
<font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/><a NAME="E6:11_2"/><b>then </b>
<font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><a NAME="E7:11_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">l</font> </span>}</span> 
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">i</font></span>)</span> <b>by </b><i><a class="ref" href="waybel_9.html#T12">WAYBEL_9:12</a></i>;<br/><font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">y</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="waybel_9.html#T16">WAYBEL_9:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">i</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">z</font>
<b>by </b><i>, <a class="ref" href="waybel_9.html#T16">WAYBEL_9:16</a></i>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">i</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>

;<br/><b>hence </b><a NAME="E10:11_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E21:11"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div>
