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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<a NAME="E1:53"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="filter_0.html#K5" target="_self">"/\"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="filter_0.html#K5" target="_self">"/\"</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="filter_0.html#T46" target="_self">Th46</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>thus </b><a NAME="E1:53_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="filter_0.html#K5" target="_self">"/\"</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:53_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:53_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="filter_0.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E5:53_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="ref" href="lattices.html#T22">LATTICES:22</a></i>;<br/>
<a NAME="E6:53_1_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="filter_0.html#E4:53_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="filter_0.html#T9" target="_self">Th9</a>, <a class="ref" href="lattices.html#T21">LATTICES:21</a></i>;<br/>
<b>hence </b><a NAME="E7:53_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="filter_0.html#K5" target="_self">"/\"</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="filter_0.html#E1:53_1_1"><i><font color="Green">E38</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:53"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="filter_0.html#K5" target="_self">"/\"</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="filter_0.html#K5" target="_self">"/\"</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="filter_0.html#E1:53"><i><font color="Green">E37</font></i></a></i>;<br/>


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