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

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="lattice3.html#K2">LattRel</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="lattice3.html#K2">LattRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E1:31_1_1"/>
<span class="p1">(<span class="default"><a href="lattice3.html#K2">LattRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">c<sub>2</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">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><a href="lattice3.html#K2">LattRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="relset_1.html#D1">RELSET_1:def 1</a></i>;<br/>
<a NAME="E3:31_1_1"/>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="lattice3.html#K2">LattRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="filter_1.html#T33">FILTER_1:33</a></i>;<br/>
<a NAME="E4:31_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
(  <a href="lattice3.html#K2">LattRel</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R1">is_reflexive_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="lattice3.html#K2">LattRel</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R4">is_antisymmetric_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="lattice3.html#K2">LattRel</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R8">is_transitive_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><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/>
<a NAME="E5:31_1_1"/><i><font color="Green">E23</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:31_1_1_1"/>
 <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="wellord1.html#T20">WELLORD1:20</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">c<sub>7</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="E2:31_1_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E3:31_1_1_1"/><b>then </b><i><font color="Green">E25</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</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/>
<a NAME="E4:31_1_1_1"/><i><font color="Green">E26</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2">LattRel</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="knaster.html#E4:31_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="knaster.html#E2:31_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="relat_2.html#D1">RELAT_2:def 1</a></i>;<br/>
<a NAME="E5:31_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="knaster.html#E3:31_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="knaster.html#E4:31_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:31_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:31_1_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="relat_2.html#V1">reflexive</a>
 
<b>by </b><i><a class="ref" href="wellord1.html#T22">WELLORD1:22</a></i>;<br/>
<a NAME="E7:31_1_1"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="knaster.html#E5:31_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a></i>;<br/>
<a NAME="E8:31_1_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="relat_2.html#V4">antisymmetric</a>
 
<b>by </b><i><a class="ref" href="wellord1.html#T25">WELLORD1:25</a></i>;<br/>
<a NAME="E9:31_1_1"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="relat_2.html#R4">is_antisymmetric_in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="knaster.html#E5:31_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="relat_2.html#D12">RELAT_2:def 12</a></i>;<br/>
<a NAME="E10:31_1_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="relat_2.html#V8">transitive</a>
 
<b>by </b><i><a class="ref" href="wellord1.html#T24">WELLORD1:24</a></i>;<br/>
<a NAME="E11:31_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="relat_2.html#R8">is_transitive_in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="knaster.html#E5:31_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/>
<a NAME="E12:31_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="knaster.html#E7:31_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="orders_1.html#T98">ORDERS_1:98</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="orders_1.html#NM2">Order</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a>, <a class="ref" href="wellord1.html#T22">WELLORD1:22</a>, <a class="ref" href="wellord1.html#T24">WELLORD1:24</a>, <a class="ref" href="wellord1.html#T25">WELLORD1:25</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> #);<br/>
<a NAME="E14:31_1_1"/><i><font color="Green">E27</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) is <a href="lattice3.html#V1">with_suprema</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D10">LATTICE3:def 10</a><br/>

<a NAME="E1:31_1_1_2"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</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>13</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="E4:31_1_1_2"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>13</sub></font> &amp; ( 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> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>13</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="knaster.html#D8" target="_self">Def8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) <b>by </b><i><a class="txt" href="knaster.html#E4:31_1_1_2"><i><font color="Green">E28</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>14</sub></font>
;<br/>

<a NAME="E6:31_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2">LattRel</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2">LattRel</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="knaster.html#E4:31_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="filter_1.html#T32">FILTER_1:32</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E7:31_1_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E8:31_1_1_2"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> #);<br/>

<b>assume </b><a NAME="E9:31_1_1_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> )
 ;<br/>

<a NAME="E10:31_1_1_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E12:31_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="knaster.html#E9:31_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E13:31_1_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2">LattRel</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2">LattRel</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E14:31_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>16</sub></font> )
 
<b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>
<a NAME="E15:31_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="knaster.html#E4:31_1_1_2"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E16:31_1_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2">LattRel</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="knaster.html#E4:31_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="filter_1.html#T32">FILTER_1:32</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E17:31_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E18:31_1_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:31_1_1"/><i><font color="Green">E28</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) is <a href="lattice3.html#V2">with_infima</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D11">LATTICE3:def 11</a><br/>

<a NAME="E1:31_1_1_3"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</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>13</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="E4:31_1_1_3"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>12</sub></font> &amp; ( 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> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>12</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>13</sub></font> ) )
 <b>by </b><i><a class="ref" href="knaster.html#D9" target="_self">Def9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) <b>by </b><i><a class="txt" href="knaster.html#E4:31_1_1_3"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>14</sub></font>
;<br/>

<a NAME="E6:31_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2">LattRel</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2">LattRel</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="knaster.html#E4:31_1_1_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="filter_1.html#T32">FILTER_1:32</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E7:31_1_1_3"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E8:31_1_1_3"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> #);<br/>

<b>assume </b><a NAME="E9:31_1_1_3"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/>

<a NAME="E10:31_1_1_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E12:31_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="knaster.html#E9:31_1_1_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E13:31_1_1_3"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2">LattRel</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2">LattRel</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E14:31_1_1_3"/><b>then </b>
( <font color="Maroon">c<sub>16</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>
<a NAME="E15:31_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="knaster.html#E4:31_1_1_3"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E16:31_1_1_3"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2">LattRel</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="knaster.html#E4:31_1_1_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="filter_1.html#T32">FILTER_1:32</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E17:31_1_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E18:31_1_1_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:31_1_1"/><i><font color="Green">E29</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) is   <a href="orders_2.html#NM1">Poset</a>
 
<b>by </b><i><a class="txt" href="knaster.html#E7:31_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="knaster.html#E9:31_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="knaster.html#E11:31_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="orders_2.html#D4">ORDERS_2:def 4</a>, <a class="ref" href="orders_2.html#D5">ORDERS_2:def 5</a>, <a class="ref" href="orders_2.html#D6">ORDERS_2:def 6</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>9</sub></font> =  <a href="lattice3.html#K14">latt</a> <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> #);<br/>

<a NAME="E17:31_1_1"/><i><font color="Green">E30</font></i>: 
 <a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="lattice3.html#K2">LattRel</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> #)
 
;<br/>
<a NAME="E18:31_1_1"/><i><font color="Green">E31</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="knaster.html#E14:31_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="knaster.html#E15:31_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="knaster.html#E16:31_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="lattice3.html#D15">LATTICE3:def 15</a></i>;<br/>
<b>thus </b><a NAME="E19:31_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="knaster.html#E14:31_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="knaster.html#E15:31_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="knaster.html#E16:31_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="knaster.html#E17:31_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="lattice3.html#D15">LATTICE3:def 15</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>9</sub></font>;<br/>

<a NAME="E20:31_1_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="knaster.html#E18:31_1_1"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>13</sub></font>
;<br/>

<b>thus </b><a NAME="E22:31_1_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_1_1_4"/>
<font color="Maroon">c<sub>10</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><a NAME="E2:31_1_1_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="knaster.html#E18:31_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/><a NAME="E3:31_1_1_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2">LattRel</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E4:31_1_1_4"/>
<font color="Maroon">c<sub>12</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>13</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="E24:31_1_1"/><i><font color="Green">E32</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="knaster.html#E18:31_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>assume </b><a NAME="E25:31_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<a NAME="E26:31_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2">LattRel</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>
<a NAME="E27:31_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2">LattRel</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="knaster.html#E18:31_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="knaster.html#E24:31_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E28:31_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>


</div>
