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

<a NAME="E1:10_1_1"/><i><font color="Green">E6</font></i>: 
 <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="filter_1.html#K9">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="filter_1.html#K9">]</a></span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>2</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="filter_1.html#D8">FILTER_1:def 8</a></i>;<br/>
<a NAME="E2:10_1_1"/>
 <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>)</span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</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="E1:10_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:10_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="filter_1.html#K9">]</a></span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="lattice3.html#E1:10_1_1"><i><font color="Green">E6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:10_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>)</span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> =  <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font> as   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <b>by </b><i><a class="ref" href="relset_1.html#D1">RELSET_1:def 1</a></i>;<br/>
<a NAME="E4:10_1_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D1">RELAT_2:def 1</a><br/>

<b>assume </b><a NAME="E1:10_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>)
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:10_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>4</sub></font>
 
;<br/>
<b>hence </b><a NAME="E4:10_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:10_1_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="relat_2.html#R4">is_antisymmetric_in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D4">RELAT_2:def 4</a><br/>

<b>assume </b><a NAME="E1:10_1_1_3"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>) &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>) )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>assume </b><a NAME="E3:10_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E4:10_1_1_3"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>
<b>hence </b><a NAME="E5:10_1_1_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="lattices.html#T26">LATTICES:26</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:10_1_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="relat_2.html#R8">is_transitive_in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <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="relat_2.html#D8">RELAT_2:def 8</a><br/>

<b>assume </b><a NAME="E1:10_1_1_4"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>) &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>) )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <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> ;<br/>
<b>assume </b><a NAME="E3:10_1_1_4"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E4:10_1_1_4"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>
<a NAME="E5:10_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="lattices.html#T25">LATTICES:25</a></i>;<br/>
<b>hence </b><a NAME="E6:10_1_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:10_1_1"/><i><font color="Green">E10</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>)
 
<b>by </b><i><a class="txt" href="lattice3.html#E4:10_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="orders_1.html#T98">ORDERS_1:98</a></i>;<br/>
<a NAME="E8:10_1_1"/>
 <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>)
 
<b>by </b><i><a class="txt" href="lattice3.html#E4:10_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="orders_1.html#T98">ORDERS_1:98</a></i>;<br/>
<b>hence </b><a NAME="E9:10_1_1"/>
 <a href="filter_1.html#K8">LattRel</a> <font color="Maroon">c<sub>1</sub></font> is   <a href="orders_1.html#NM2">Order</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="lattice3.html#E4:10_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="lattice3.html#E5:10_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="lattice3.html#E6:10_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="lattice3.html#E7:10_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a>, <a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a>, <a class="ref" href="relat_2.html#D12">RELAT_2:def 12</a>, <a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/>


</div>
