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<span class="kw">set </span><font color="Maroon" title="c2">p</font> =  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font>;<br/>
<a NAME="E1:57_1_1"/>
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="latticea.html#T1" target="_self" onmouseover="rs('latticea/T1')" onmouseout="rh()">TopBot</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">I</font> being   <a href="filter_2.html#NM1" title="FILTER_2:NM.1">Ideal</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E3:57_1_1"/><span class="lab"><font color="Green" title="E22">A0</font></span>: 
(  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Maroon" title="c3">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="ref" href="filter_2.html#T35" onmouseover="rs('filter_2/T35')" onmouseout="rh()">FILTER_2:35</a></span>;<br/>
<a NAME="E4:57_1_1"/><span class="lab"><font color="Green" title="E23">a2</font></span>: 
<font color="Maroon" title="c3">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>, <a class="txt" href="#E3:57_1_1"><span class="lab"><font color="Green" title="E22">A0</font></span></a></span>;<br/>
<a NAME="E5:57_1_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="c3">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="c3">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="57_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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="c4">G</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  &amp; <font color="Maroon" title="c3">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">G</font> implies <font color="Maroon" title="c3">I</font> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <font color="Maroon" title="c4">G</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:57_1_1_1"/><span class="lab"><font color="Green" title="E24">B1</font></span>: 
( <font color="Maroon" title="c4">G</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  &amp; <font color="Maroon" title="c3">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">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="c3">I</font> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <font color="Maroon" title="c4">G</font></span><br/>

<a NAME="E2:57_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">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="E3:57_1_1_1"/>
<font color="Maroon" title="c3">I</font> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <font color="Maroon" title="c4">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="#E3:57_1_1"><span class="lab"><font color="Green" title="E22">A0</font></span></a>, <a class="txt" href="#E1:57_1_1_1"><span class="lab"><font color="Green" title="E24">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>
<a NAME="E6:57_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">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="#E4:57_1_1"><span class="lab"><font color="Green" title="E23">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>;<br/>
<a NAME="E7:57_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">I</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latticea.html#K8" title="LATTICEA:func.8">Spectrum</a> <font color="Maroon" title="c1">L</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E8:57_1_1"/>
 not  <a href="latticea.html#K8" title="LATTICEA:func.8">Spectrum</a> <font color="Maroon" title="c1">L</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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"> verum</span><br/>


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