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

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">J</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> =  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E1:29_1_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">J</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</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>consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">J</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:29_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">J</font> holds  <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S8">FUNCT_2:sch 8</a>();<br/></i>
<b>set </b><font color="Maroon">M</font> =  <a href="waybel_0.html#G1">NetStr</a>(# <span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">J</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="finsub_1.html#K5">Fin</a> <font color="Maroon">J</font></span>)</span></span>)</span>,<font color="Maroon">g</font> #);<br/>
<a NAME="E4:29_1_1"/>
 <a href="waybel_0.html#G1">NetStr</a>(# <span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">J</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="finsub_1.html#K5">Fin</a> <font color="Maroon">J</font></span>)</span></span>)</span>,<font color="Maroon">g</font> #) is <a href="waybel_0.html#V7">directed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="waybel_0.html#G1">NetStr</a>(# <span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">J</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="finsub_1.html#K5">Fin</a> <font color="Maroon">J</font></span>)</span></span>)</span>,<font color="Maroon">g</font> #), <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="waybel_0.html#G1">NetStr</a>(# <span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">J</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="finsub_1.html#K5">Fin</a> <font color="Maroon">J</font></span>)</span></span>)</span>,<font color="Maroon">g</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a>,<a class="ref" href="waybel_0.html#D6">WAYBEL_0:def 6</a><br/>

<b>assume </b><a NAME="E1:29_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <a href="waybel_0.html#G1">NetStr</a>(# <span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">J</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="finsub_1.html#K5">Fin</a> <font color="Maroon">J</font></span>)</span></span>)</span>,<font color="Maroon">g</font> #) &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <a href="waybel_0.html#G1">NetStr</a>(# <span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">J</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="finsub_1.html#K5">Fin</a> <font color="Maroon">J</font></span>)</span></span>)</span>,<font color="Maroon">g</font> #) )
 ;<br/>

<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font>, <font color="Maroon">y1</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">J</font> ;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">x1</font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">y1</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="waybel_0.html#G1">NetStr</a>(# <span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">J</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="finsub_1.html#K5">Fin</a> <font color="Maroon">J</font></span>)</span></span>)</span>,<font color="Maroon">g</font> #) ;<br/>
<b>take </b>
<font color="Maroon">z</font>
;<br/>

<a NAME="E4:29_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="waybel_0.html#G1">NetStr</a>(# <span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">J</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="finsub_1.html#K5">Fin</a> <font color="Maroon">J</font></span>)</span></span>)</span>,<font color="Maroon">g</font> #)
 
;<br/>
<b>hence </b><a NAME="E5:29_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <a href="waybel_0.html#G1">NetStr</a>(# <span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">J</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="finsub_1.html#K5">Fin</a> <font color="Maroon">J</font></span>)</span></span>)</span>,<font color="Maroon">g</font> #)
 <b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>

<a NAME="E6:29_1_1_1"/><i><font color="Green">E34</font></i>: 
 <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">J</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">J</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="finsub_1.html#K5">Fin</a> <font color="Maroon">J</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 reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">x</font>, <font color="Maroon">y2</font> = <font color="Maroon">y</font>, <font color="Maroon">z1</font> = <font color="Maroon">z</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="finsub_1.html#K5">Fin</a> <font color="Maroon">J</font></span>)</span></span>)</span> ;<br/>
<a NAME="E8:29_1_1_1"/>
( <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">z</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E9:29_1_1_1"/><b>then </b>
( <font color="Maroon">x2</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z1</font> &amp; <font color="Maroon">y2</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z1</font> )
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>
<b>hence </b><a NAME="E10:29_1_1_1"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">M</font> =  <a href="waybel_0.html#G1">NetStr</a>(# <span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">J</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="finsub_1.html#K5">Fin</a> <font color="Maroon">J</font></span>)</span></span>)</span>,<font color="Maroon">g</font> #) as   <a href="waybel_0.html#NM1">prenet</a> of <font color="Maroon">L</font> ;<br/>
<b>take </b>
<font color="Maroon">M</font>
;<br/>

<b>thus </b><a NAME="E6:29_1_1"/>
 ex <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">J</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> st <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">J</font> holds <br/> ( <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">x</font></span>)</span> &amp; <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#G1">NetStr</a>(# <span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">J</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="finsub_1.html#K5">Fin</a> <font color="Maroon">J</font></span>)</span></span>)</span>,<font color="Olive">g</font> #) )
 <b>by </b><i/>;<br/>


</div>
