<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattice3.html#V4" target="_self">complete</a> <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> ;<br/>
<a NAME="E1:85"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  <a href="lattice3.html#R4" target="_self">is_less_than</a>  <a href="lattice3.html#K15" target="_self">"\/"</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17" target="_self">LATTICE3:def 17</a><br/>

<b>assume </b><a NAME="E1:85_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:85_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>
<a NAME="E4:85_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="lattice3.html#E3:85_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E5:85_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#R1">[=</a>  <a href="lattice3.html#K15" target="_self">"\/"</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="lattice3.html#E3:85_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="lattice3.html#T46" target="_self">Th46</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:85"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="lattice3.html#K15" target="_self">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> ,<font color="Maroon">c<sub>1</sub></font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K15" target="_self">"\/"</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D21" target="_self">Def21</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <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> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Olive">b<sub>3</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  ) </span>}</span> ;<br/>
<b>assume </b><a NAME="E3:85"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E4:85"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <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> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Olive">b<sub>3</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</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="E1:85_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:85_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="lattice3.html#E3:85"><i><font color="Green">E50</font></i></a>, <a class="txt" href="lattice3.html#E3:85_2"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="lattice3.html#E3:85"><i><font color="Green">E50</font></i></a>, <a class="txt" href="lattice3.html#E3:85_2"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E6:85_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  )
 
<b>by </b><i><a class="txt" href="lattice3.html#E3:85_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="lattice3.html#T38" target="_self">Th38</a></i>;<br/>
<b>hence </b><a NAME="E7:85_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <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> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Olive">b<sub>3</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  ) </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:85"/><b>then </b>
 <a href="lattice3.html#K15" target="_self">"\/"</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K15" target="_self">"\/"</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <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> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Olive">b<sub>3</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  ) </span>}</span> ,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T46" target="_self">Th46</a></i>;<br/>
<a NAME="E6:85"/><b>then </b>
 <a href="lattice3.html#K15" target="_self">"\/"</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K15" target="_self">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> ,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T47" target="_self">Th47</a></i>;<br/>
<b>hence </b><a NAME="E7:85"/>
 <a href="lattice3.html#K15" target="_self">"\/"</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15" target="_self">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> ,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="lattice3.html#E2:85"><i><font color="Green">E49</font></i></a>, <a class="ref" href="lattices.html#T26">LATTICES:26</a></i>;<br/>


</div>
