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

<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">I</font> being   <a href="filter_2.html#M2" title="FILTER_2:mode.2">Ideal</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> holds <br/> <a href="filter_2.html#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<font color="Olive" title="b1">I</font> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a></span><br/><b>let </b><font color="Maroon" title="c2">I</font> be    <a href="filter_2.html#M2" title="FILTER_2:mode.2">Ideal</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> implies  <a href="filter_2.html#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">I</font> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> )</span><br/>



<b>given </b><font color="Maroon" title="c3">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><a NAME="E2:106"/><i><font color="Green" title="E43">A1</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Maroon" title="c3">c</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Olive" title="b1">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D13" title="LATTICES:def.13">LATTICES:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="filter_2.html#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">I</font> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a></span><br/>

<b>consider </b><font color="Maroon" title="c4">b'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="filter_2.html#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">I</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">b</font> = <font color="Maroon" title="c4">b'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="filter_2.html#T69" target="_self" title="FILTER_2:th.69">Th69</a></i>;<br/>
<a NAME="E5:106"/>
( <font color="Maroon">H<sub>1</sub></font>( <a href="filter_2.html#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">I</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Maroon" title="c3">c</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> )
 
<b>by </b><i><a class="txt" href="#E2:106"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="filter_2.html#T73" target="_self" title="FILTER_2:th.73">Th73</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">c'</font> = <font color="Maroon" title="c3">c</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="filter_2.html#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">I</font></span>)</span> <b>by </b><i><a class="ref" href="filter_2.html#T23" target="_self" title="FILTER_2:th.23">Th23</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">c'</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D13" title="LATTICES:def.13">LATTICES:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="filter_2.html#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">I</font></span>)</span> holds <br/> ( <font color="Maroon" title="c6">c'</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c'</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c6">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c'</font> )</span><br/>

<b>let </b><font color="Maroon" title="c7">a'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="filter_2.html#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">I</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">c'</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c7">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c'</font> &amp; <font color="Maroon" title="c7">a'</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c6">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c'</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c8">a</font> = <font color="Maroon" title="c7">a'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="filter_2.html#T69" target="_self" title="FILTER_2:th.69">Th69</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c6">c'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">a'</font> = 
<font color="Maroon" title="c3">c</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">a</font>
<b>by </b><i><a class="ref" href="filter_2.html#T74" target="_self" title="FILTER_2:th.74">Th74</a></i>
<br/>.= 
<font color="Maroon" title="c6">c'</font>
<b>by </b><i><a class="txt" href="#E2:106"><i><font color="Green" title="E43">A1</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">a'</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c6">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c'</font></span><br/>

<b>hence </b><a NAME="E9:106"/>
<font color="Maroon" title="c7">a'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
