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<b>let </b><font color="Maroon">T</font> be  <a href="abcmiz_0.html#V1" target="_self">Noetherian</a> <a href="waybel_0.html#NM7">sup-Semilattice</a>;<br/><b>let </b><font color="Maroon">I</font> be   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">T</font>;<br/>



<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E2:11"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>and </b><br/><a NAME="E3:11"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/> not <font color="Maroon">a</font> <a href="orders_2.html#R2">&lt;</a> <font color="Olive">b</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:11"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">I</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:11_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<a NAME="E2:11_1"/><b>then </b>
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T40">WAYBEL_0:40</a></i>;<br/>
<a NAME="E3:11_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">d</font> &amp; not <font color="Maroon">a</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">d</font> )
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<a NAME="E4:11_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/>
<b>hence </b><a NAME="E5:11_1"/>
<font color="Maroon">d</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:11"/>
for <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>  st <font color="Maroon">I</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">c</font> holds <br/><font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">c</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E6:11"/>
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">I</font>,<font color="Maroon">T</font> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 <b>by </b><i>, , <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/>


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