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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> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  is  <a href="filter_0.html#V1" title="FILTER_0:attr.1">being_ultrafilter</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> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  is  <a href="filter_0.html#V1" title="FILTER_0:attr.1">being_ultrafilter</a>  )</span><br/>



<span class="kw">thus </span><a NAME="E1:45"/>
( <font color="Maroon" title="c2">I</font> is  <a href="filter_2.html#V1" title="FILTER_2:attr.1">max-ideal</a>  implies <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  is  <a href="filter_0.html#V1" title="FILTER_0:attr.1">being_ultrafilter</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> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  is  <a href="filter_0.html#V1" title="FILTER_0:attr.1">being_ultrafilter</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><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:45_1"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
<font color="Maroon" title="c2">I</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">L</font>)
 <span class="kw">and </span><br/><a NAME="E2:45_1"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
 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> <font color="Maroon">H<sub>1</sub></font>(<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>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="filter_2.html#D8" target="_self" onmouseover="rs('filter_2/D8')" onmouseout="rh()">FILTER_2:def 8</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> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  is  <a href="filter_0.html#V1" title="FILTER_0:attr.1">being_ultrafilter</a> </span><br/>

<span class="kw">thus </span><a NAME="E3:45_1"/>
<font color="Maroon" title="c2">I</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1"><span class="lab"><font color="Green" title="E17">A1</font></span></a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="filter_0.html#D3" onmouseover="rs('filter_0/D3')" onmouseout="rh()">FILTER_0:def 3</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">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a></span>)</span>  holds <br/>(  not <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a></span>)</span> or <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">F</font> be   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a></span>)</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"> (  not <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> or <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a></span>)</span> or <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> )</span><br/>

<a NAME="E4:45_1"/>
 <a href="filter_2.html#K4" title="FILTER_2:func.4">.:</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E5:45_1"/>
(  not <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> or <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a></span>)</span> or <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1"><span class="lab"><font color="Green" title="E18">A2</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">assume </span><span class="kw">that </span><br/><a NAME="E2:45"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
<font color="Maroon" title="c2">I</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> )
 <span class="kw">and </span><br/><a NAME="E3:45"/><span class="lab"><font color="Green" title="E18">A4</font></span>: 
 for <font color="Olive" title="b1">F</font> being   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a></span>)</span>  st <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b1">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> ) holds <br/><font color="Maroon" title="c2">I</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <font color="Olive" title="b1">F</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="filter_0.html#D3" onmouseover="rs('filter_0/D3')" onmouseout="rh()">FILTER_0:def 3</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/>

<span class="kw">thus </span><a NAME="E4:45"/>
<font color="Maroon" title="c2">I</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">L</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E17">A3</font></span></a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="filter_2.html#D8" target="_self" onmouseover="rs('filter_2/D8')" onmouseout="rh()">FILTER_2:def 8</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">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></span><br/>

<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/>

<a NAME="E5:45"/>
<font color="Maroon" title="c3">J</font> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">J</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E6:45"/>
( <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E18">A4</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/>


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