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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"> ( <font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a>  implies  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>  st <font color="Olive" title="b1">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> holds <br/> ex <font color="Olive" title="b2">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 <br/>( <font color="Olive" title="b1">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">J</font> &amp; <font color="Olive" title="b2">J</font> is  <a href="filter_2.html#V1" title="FILTER_2:attr.1">max-ideal</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:46"/>
<font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</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>  st <font color="Olive" title="b1">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> holds <br/> ex <font color="Olive" title="b2">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 <br/>( <font color="Olive" title="b1">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">J</font> &amp; <font color="Olive" title="b2">J</font> is  <a href="filter_2.html#V1" title="FILTER_2:attr.1">max-ideal</a>  )</span><br/>

<a NAME="E2:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A1</font></span>: 
<font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  is  <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice2.html#T49" onmouseover="rs('lattice2/T49')" onmouseout="rh()">LATTICE2:49</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> <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  ex <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 <br/>( <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> is  <a href="filter_2.html#V1" title="FILTER_2:attr.1">max-ideal</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E3:46"/>
<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>)
 ; <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">  ex <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 <br/>( <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> is  <a href="filter_2.html#V1" title="FILTER_2:attr.1">max-ideal</a>  )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">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><span class="kw"> such that </span><br/><a NAME="E5:46"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
( <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> &amp; <font color="Maroon" title="c3">F</font> is  <a href="filter_0.html#V1" title="FILTER_0:attr.1">being_ultrafilter</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="filter_0.html#T18" onmouseover="rs('filter_0/T18')" onmouseout="rh()">FILTER_0:18</a></span>;<br/>
<span class="kw">take </span>
 <a href="filter_2.html#K4" title="FILTER_2:func.4">.:</a> <font color="Maroon" title="c3">F</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>  <a href="filter_2.html#K4" title="FILTER_2:func.4">.:</a> <font color="Maroon" title="c3">F</font> &amp;  <a href="filter_2.html#K4" title="FILTER_2:func.4">.:</a> <font color="Maroon" title="c3">F</font> is  <a href="filter_2.html#V1" title="FILTER_2:attr.1">max-ideal</a>  )</span><br/>

<a NAME="E6:46"/>
<span class="p1">(<span class="default"><a href="filter_2.html#K4" title="FILTER_2:func.4">.:</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="filter_2.html#K4" title="FILTER_2:func.4">.:</a> <font color="Maroon" title="c3">F</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E7:46"/>
( <font color="Maroon" title="c2">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="filter_2.html#K4" title="FILTER_2:func.4">.:</a> <font color="Maroon" title="c3">F</font> &amp;  <a href="filter_2.html#K4" title="FILTER_2:func.4">.:</a> <font color="Maroon" title="c3">F</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="#E5:46"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="filter_2.html#T32" target="_self" onmouseover="rs('filter_2/T32')" onmouseout="rh()">Th32</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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