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

<b>set </b><font color="Maroon">P</font> =  <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">S</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">R</font> =  <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span>;<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">J</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">J</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Olive">J</font> );<br/>
<a NAME="E1:83_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">I</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">S</font></span>)</span></span>)</span>  ex <font color="Olive">K</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">I</font>,<font color="Olive">K</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">I</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">S</font></span>)</span></span>)</span>;<br/>

<a NAME="E1:83_1_1_1"/>
<font color="Maroon">I</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#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">S</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E2:83_1_1_1"/><b>then </b>
<font color="Maroon">I</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K7">Ids</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E3:83_1_1_1"/><b>then </b>
<font color="Maroon">I</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">S</font> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel_0.html#D23">WAYBEL_0:def 23</a></i>;<br/>
<b>then consider </b><font color="Maroon">J</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E5:83_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">J</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">J</font> = <font color="Maroon">J</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="yellow_2.html#T7">YELLOW_2:7</a></i>;<br/>
<a NAME="E7:83_1_1_1"/>
 <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">J</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : verum </span>}</span> 
 
;<br/>
<a NAME="E8:83_1_1_1"/><b>then </b>
 <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">J</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D23">WAYBEL_0:def 23</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">K</font> =  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">J</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<b>take </b>
<font color="Maroon">K</font>
;<br/>

<b>take </b>
<font color="Maroon">J</font>
;<br/>

<b>thus </b><a NAME="E10:83_1_1_1"/>
( <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> &amp; <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">J</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:83_1_1"/>
 ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">S</font></span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span> st <br/>for <font color="Olive">I</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">S</font></span>)</span></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">I</font>,<font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">I</font>]
 
<b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="ref" href="waybel23.html#T3" target="_self">Th3</a>);<br/></i>
<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">S</font></span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:83_1_1"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">I</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">S</font></span>)</span></span>)</span>  ex <font color="Olive">J</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">J</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">I</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Olive">J</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">S</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<a NAME="E6:83_1_1"/>
for <font color="Olive">I</font> being  <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">S</font>  ex <font color="Olive">J</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">J</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">I</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Olive">J</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">I</font> be   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">S</font>;<br/>

<a NAME="E1:83_1_1_2"/>
<font color="Maroon">I</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">S</font> : verum </span>}</span> 
 
;<br/>
<a NAME="E2:83_1_1_2"/><b>then </b>
<font color="Maroon">I</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">S</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">S</font></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="waybel_0.html#D23">WAYBEL_0:def 23</a></i>;<br/>
<a NAME="E3:83_1_1_2"/><b>then </b>
<font color="Maroon">I</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#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">S</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D1">YELLOW_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">J</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:83_1_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font>
 <b>and </b><br/><a NAME="E6:83_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">J</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">J</font> = <font color="Maroon">J</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/>
<b>take </b>
<font color="Maroon">J</font>
;<br/>

<b>thus </b><a NAME="E8:83_1_1_2"/>
( <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">J</font> )
 <b>by </b><i>, <a class="ref" href="waybel23.html#T5" target="_self">Th5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:83_1_1"/>
for <font color="Olive">I</font> being  <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">S</font>  ex <font color="Olive">J</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">J</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">I</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Olive">J</font> )
 ;<br/>


</div>
