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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="lattices.html#NM1">Lattice</a>;<br/>



<b>thus </b><a NAME="E1:60"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V14">upper-bounded</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="lattices.html#V14">upper-bounded</a> implies <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="filter_1.html#K7" target="_self">:]</a></span> is <a href="lattices.html#V14">upper-bounded</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E1:60_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D14">LATTICES:def 14</a><br/>

<b>given </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><a NAME="E2:60_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D14">LATTICES:def 14</a><br/>

<b>take </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D14">LATTICES:def 14</a><br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="filter_1.html#K7" target="_self">:]</a></span>;<br/>

<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:60_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span>
<b>by </b><i><a class="txt" href="filter_1.html#E4:60_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="filter_1.html#T36" target="_self">Th36</a></i>
<br/>.= 
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span>
<b>by </b><i><a class="txt" href="filter_1.html#E1:60_1"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="filter_1.html#E2:60_1"><i><font color="Green">E40</font></i></a></i>
;<br/>

<b>hence </b><a NAME="E6:60_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>


</div><b>end;</b></div>

<b>given </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="filter_1.html#K7" target="_self">:]</a></span><b> such that </b><a NAME="E2:60"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="filter_1.html#K7" target="_self">:]</a></span> holds <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D14">LATTICES:def 14</a><br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:60"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<b>thus </b><a NAME="E5:60"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V14">upper-bounded</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D14">LATTICES:def 14</a><br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <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>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span>
<b>by </b><i><a class="txt" href="filter_1.html#E2:60"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span>
<b>by </b><i><a class="txt" href="filter_1.html#E4:60"><i><font color="Green">E40</font></i></a>, <a class="ref" href="filter_1.html#T36" target="_self">Th36</a></i>
;<br/>
<b>hence </b><a NAME="E3:60_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="filter_1.html#E4:60"><i><font color="Green">E40</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>


</div><b>end;</b></div>

<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D14">LATTICES:def 14</a><br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="filter_1.html#K9" target="_self">]</a></span>
<b>by </b><i><a class="txt" href="filter_1.html#E2:60"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span>
<b>by </b><i><a class="txt" href="filter_1.html#E4:60"><i><font color="Green">E40</font></i></a>, <a class="ref" href="filter_1.html#T36" target="_self">Th36</a></i>
;<br/>
<b>hence </b><a NAME="E8:60"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="filter_1.html#E4:60"><i><font color="Green">E40</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>


</div>
