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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a>  <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">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st  not <font color="Olive" title="b1">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">b</font> holds <br/> ex <font color="Olive" title="b3">P</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="b3">P</font> is  <a href="filter_2.html#V2" title="FILTER_2:attr.2">prime</a>  &amp;  not <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">P</font> &amp; <font color="Olive" title="b2">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">P</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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"> (  not <font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">b</font> implies  ex <font color="Olive" title="b1">P</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">P</font> is  <a href="filter_2.html#V2" title="FILTER_2:attr.2">prime</a>  &amp;  not <font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c3">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">P</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:35"/><span class="lab"><font color="Green" title="E17">AA</font></span>: 
 not <font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">b</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">P</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">P</font> is  <a href="filter_2.html#V2" title="FILTER_2:attr.2">prime</a>  &amp;  not <font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c3">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">P</font> )</span><br/>

<a NAME="E2:35"/><span class="lab"><font color="Green" title="E18">ZZ</font></span>: 
<span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="35_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:35_1"/>
<span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a> <span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</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"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:35_1"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a></span>;<br/>
<a NAME="E5:35_1"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="filter_2.html#T28" onmouseover="rs('filter_2/T28')" onmouseout="rh()">FILTER_2:28</a></span>;<br/>
<a NAME="E6:35_1"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="filter_0.html#T15" onmouseover="rs('filter_0/T15')" onmouseout="rh()">FILTER_0:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:35_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E17">AA</font></span></a>, <a class="txt" href="#E5:35_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="lattices.html#T7" onmouseover="rs('lattices/T7')" onmouseout="rh()">LATTICES:7</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">set </span><font color="Maroon" title="c4">I1</font> = <span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">F1</font> = <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">P</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="E4:35"/><span class="lab"><font color="Green" title="E19">B1</font></span>: 
( <font color="Maroon" title="c6">P</font> is  <a href="filter_2.html#V2" title="FILTER_2:attr.2">prime</a>  &amp; <span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c6">P</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="latticea.html#T19" target="_self" onmouseover="rs('latticea/T19')" onmouseout="rh()">Th15</a>, <a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E18">ZZ</font></span></a></span>;<br/>
<a NAME="E5:35"/><span class="lab"><font color="Green" title="E20">B2</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35"><span class="lab"><font color="Green" title="E19">B1</font></span></a>, <a class="ref" href="filter_2.html#T28" onmouseover="rs('filter_2/T28')" onmouseout="rh()">FILTER_2:28</a></span>;<br/>
<a NAME="E6:35"/>
<font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span>
 
;<br/>
<a NAME="E7:35"/><span class="kw">then </span>
 not <font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35"><span class="lab"><font color="Green" title="E19">B1</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:35"/>
 ex <font color="Olive" title="b1">P</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">P</font> is  <a href="filter_2.html#V2" title="FILTER_2:attr.2">prime</a>  &amp;  not <font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c3">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">P</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35"><span class="lab"><font color="Green" title="E19">B1</font></span></a>, <a class="txt" href="#E5:35"><span class="lab"><font color="Green" title="E20">B2</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>
