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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</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="filter_2.html#NM1" title="FILTER_2:NM.1">Ideal</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Olive" title="b1">I</font> is  <a href="filter_2.html#V1" title="FILTER_2:attr.1">max-ideal</a>  iff <font color="Olive" title="b1">I</font> is  <a href="latticea.html#V3" title="LATTICEA:attr.3">maximal</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">I</font> be   <a href="filter_2.html#NM1" title="FILTER_2:NM.1">Ideal</a> of <font color="Maroon" title="c1">L</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="c2">I</font> is  <a href="filter_2.html#V1" title="FILTER_2:attr.1">max-ideal</a>  iff <font color="Maroon" title="c2">I</font> is  <a href="latticea.html#V3" title="LATTICEA:attr.3">maximal</a>  )</span><br/>



<div><span class="kw">hereby </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"> ( <font color="Maroon" title="c2">I</font> is  <a href="latticea.html#V3" title="LATTICEA:attr.3">maximal</a>  implies <font color="Maroon" title="c2">I</font> is  <a href="filter_2.html#V1" title="FILTER_2:attr.1">max-ideal</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:21_1"/><span class="lab"><font color="Green" title="E9">A0</font></span>: 
<font color="Maroon" title="c2">I</font> is  <a href="filter_2.html#V1" title="FILTER_2:attr.1">max-ideal</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"> <font color="Maroon" title="c2">I</font> is  <a href="latticea.html#V3" title="LATTICEA:attr.3">maximal</a> </span><br/><a NAME="E2:21_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">a2</font></span>: 
<font color="Maroon" title="c2">I</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#D8" onmouseover="rs('filter_2/D8')" onmouseout="rh()">FILTER_2:def 8</a></span>;<br/><a NAME="E3:21_1"/>
 for <font color="Olive" title="b1">G</font> being   <a href="filter_2.html#NM1" title="FILTER_2:NM.1">Ideal</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">G</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  &amp; <font color="Maroon" title="c2">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">G</font> holds <br/><font color="Maroon" title="c2">I</font> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <font color="Olive" title="b1">G</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">G</font> be   <a href="filter_2.html#NM1" title="FILTER_2:NM.1">Ideal</a> of <font color="Maroon" title="c1">L</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="c3">G</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  &amp; <font color="Maroon" title="c2">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">G</font> implies <font color="Maroon" title="c2">I</font> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <font color="Maroon" title="c3">G</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_1_1"/><span class="lab"><font color="Green" title="E11">B1</font></span>: 
( <font color="Maroon" title="c3">G</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  &amp; <font color="Maroon" title="c2">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">G</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="c2">I</font> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <font color="Maroon" title="c3">G</font></span><br/>

<a NAME="E2:21_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#D6" onmouseover="rs('subset_1/D6')" onmouseout="rh()">SUBSET_1:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:21_1_1"/>
<font color="Maroon" title="c2">I</font> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <font color="Maroon" title="c3">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#D8" onmouseover="rs('filter_2/D8')" onmouseout="rh()">FILTER_2:def 8</a>, <a class="txt" href="#E1:21_1"><span class="lab"><font color="Green" title="E9">A0</font></span></a>, <a class="txt" href="#E1:21_1_1"><span class="lab"><font color="Green" title="E11">B1</font></span></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><span class="kw">hence </span><a NAME="E4:21_1"/>
<font color="Maroon" title="c2">I</font> is  <a href="latticea.html#V3" title="LATTICEA:attr.3">maximal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1"><span class="lab"><font color="Green" title="E10">a2</font></span></a>, <a class="ref" href="subset_1.html#D6" onmouseover="rs('subset_1/D6')" onmouseout="rh()">SUBSET_1:def 6</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>

<span class="kw">assume </span><a NAME="E2:21"/><span class="lab"><font color="Green" title="E9">A0</font></span>: 
<font color="Maroon" title="c2">I</font> is  <a href="latticea.html#V3" title="LATTICEA:attr.3">maximal</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"> <font color="Maroon" title="c2">I</font> is  <a href="filter_2.html#V1" title="FILTER_2:attr.1">max-ideal</a> </span><br/>

<a NAME="E3:21"/>
 for <font color="Olive" title="b1">J</font> being   <a href="filter_2.html#NM1" title="FILTER_2:NM.1">Ideal</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c2">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">J</font> &amp; <font color="Olive" title="b1">J</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> holds <br/><font color="Maroon" title="c2">I</font> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <font color="Olive" title="b1">J</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">J</font> be   <a href="filter_2.html#NM1" title="FILTER_2:NM.1">Ideal</a> of <font color="Maroon" title="c1">L</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="c2">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">J</font> &amp; <font color="Maroon" title="c3">J</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> implies <font color="Maroon" title="c2">I</font> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <font color="Maroon" title="c3">J</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_2"/><span class="lab"><font color="Green" title="E10">B1</font></span>: 
( <font color="Maroon" title="c2">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">J</font> &amp; <font color="Maroon" title="c3">J</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</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="c2">I</font> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <font color="Maroon" title="c3">J</font></span><br/>

<a NAME="E2:21_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">J</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#D6" onmouseover="rs('subset_1/D6')" onmouseout="rh()">SUBSET_1:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:21_2"/>
<font color="Maroon" title="c2">I</font> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <font color="Maroon" title="c3">J</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E9">A0</font></span></a>, <a class="txt" href="#E1:21_2"><span class="lab"><font color="Green" title="E10">B1</font></span></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>
<span class="kw">hence </span><a NAME="E4:21"/>
<font color="Maroon" title="c2">I</font> is  <a href="filter_2.html#V1" title="FILTER_2:attr.1">max-ideal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E9">A0</font></span></a>, <a class="ref" href="filter_2.html#D8" onmouseover="rs('filter_2/D8')" onmouseout="rh()">FILTER_2:def 8</a>, <a class="ref" href="subset_1.html#D6" onmouseover="rs('subset_1/D6')" onmouseout="rh()">SUBSET_1:def 6</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>
