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

<span class="kw">let </span><font color="Maroon" title="c1">T</font> be   <a href="abcmiz_0.html#V1" title="ABCMIZ_0:attr.1">Noetherian</a>  <a href="waybel_0.html#NM7" title="WAYBEL_0:NM.7">sup-Semilattice</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">I</font> being   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">T</font> holds <br/> (  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b1">I</font>,<font color="Maroon" title="c1">T</font> &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">I</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">I</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">I</font> be   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c1">T</font> &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c2">I</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">I</font> )</span><br/>



<span class="kw">consider </span><font color="Maroon" title="c3">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E2:11"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 <span class="kw">and </span><br/><a NAME="E3:11"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
 for <font color="Olive" title="b1">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">I</font> holds <br/> not <font color="Maroon" title="c3">a</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_0.html#D2" target="_self" onmouseover="rs('abcmiz_0/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<a NAME="E4:11"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
<font color="Maroon" title="c2">I</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c3">a</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="11_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattice3.html#D9" onmouseover="rs('lattice3/D9')" onmouseout="rh()">LATTICE3:def 9</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">I</font> or <font color="Maroon" title="c4">d</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:11_1"/>
<font color="Maroon" title="c4">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">d</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">a</font></span><br/>

<a NAME="E2:11_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="waybel_0.html#T40" onmouseover="rs('waybel_0/T40')" onmouseout="rh()">WAYBEL_0:40</a></span>;<br/>
<a NAME="E3:11_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A4</font></span>: 
 not <font color="Maroon" title="c3">a</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c3">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11"><span class="lab"><font color="Green" title="E5">A2</font></span></a></span>;<br/>
<a NAME="E4:11_1"/>
<font color="Maroon" title="c3">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T22" onmouseover="rs('yellow_0/T22')" onmouseout="rh()">YELLOW_0:22</a></span>;<br/>
<a NAME="E5:11_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11_1"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="ref" href="orders_2.html#D6" onmouseover="rs('orders_2/D6')" onmouseout="rh()">ORDERS_2:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:11_1"/>
<font color="Maroon" title="c4">d</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T22" onmouseover="rs('yellow_0/T22')" onmouseout="rh()">YELLOW_0:22</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:11"/>
 for <font color="Olive" title="b1">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>  st <font color="Maroon" title="c2">I</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">c</font> holds <br/><font color="Maroon" title="c3">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11"><span class="lab"><font color="Green" title="E4">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:11"/>
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c1">T</font> &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c2">I</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">I</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E4:11"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="ref" href="yellow_0.html#T30" onmouseover="rs('yellow_0/T30')" onmouseout="rh()">YELLOW_0:30</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
