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

<b>let </b><font color="Maroon">L</font> be   <a href="waybel_0.html#NM7">sup-Semilattice</a>;<br/><b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V12" target="_self">lower</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>



<b>set </b><font color="Maroon">S</font> =  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">X</font>;<br/>
<a NAME="E1:120"/><i><font color="Green">E28</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:120_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">X</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font>
 ;<br/><b>thus </b><a NAME="E2:120_1"/>
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">X</font> is <a href="yellow_0.html#V6">join-inheriting</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_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 <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D17">YELLOW_0:def 17</a><br/>

<b>assume </b><a NAME="E1:120_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">x</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_0.html#K5">subrelstr</a> <font color="Maroon">X</font></span>)</span> &amp; <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"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">X</font></span>)</span> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">L</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:120_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#D4" target="_self">Def4</a>, , </i>;<br/>
<a NAME="E4:120_1_1"/>
<font color="Maroon">z</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T8">YELLOW_0:8</a></i>;<br/>
<a NAME="E5:120_1_1"/><b>then </b>
<font color="Maroon">z</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/>
<b>hence </b><a NAME="E6:120_1_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> <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_0.html#K5">subrelstr</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_0.html#D4" target="_self">Def4</a>, , <a class="ref" href="waybel_0.html#T1" target="_self">Th1</a></i>;<br/>


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

<b>assume </b><a NAME="E3:120"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">x</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_0.html#K5">subrelstr</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Olive">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"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">X</font></span>)</span> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">L</font> holds <br/> <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K7">}</a></span> <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_0.html#K5">subrelstr</a> <font color="Maroon">X</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D17">YELLOW_0:def 17</a><br/>

<a NAME="E4:120"/>
<font color="Maroon">X</font> is <a href="waybel_0.html#V1" target="_self">directed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_2"><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 <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D1" target="_self">WAYBEL_0:def 1</a><br/>

<b>assume </b><a NAME="E1:120_2"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 ;<br/>

<b>take </b><font color="Maroon">z</font> =  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>;<br/>

<a NAME="E2:120_2"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T20">YELLOW_0:20</a>, <a class="ref" href="yellow_0.html#T41">YELLOW_0:41</a></i>;<br/>
<b>hence </b><a NAME="E3:120_2"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#D4" target="_self">Def4</a>, , , <a class="ref" href="yellow_0.html#T18">YELLOW_0:18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:120"/>
<font color="Maroon">X</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font>
 ;<br/>


</div>
