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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="orders_2.html#V4">antisymmetric</a> <a href="yellow_0.html#V3">bounded</a> <a href="lattice3.html#V1">with_suprema</a> <a href="lattice3.html#V2">with_infima</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3">:]</a></span>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:20_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="yellow10.html#R1" target="_self">is_a_complement_of</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>thus </b><a NAME="E2:20_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="yellow_3.html#K8">`1</a>  <a href="yellow10.html#R1" target="_self">is_a_complement_of</a> <font color="Maroon">c<sub>4</sub></font> <a href="yellow_3.html#K8">`1</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>proof </b></a><div class="add">

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="yellow_3.html#K8">`1</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="yellow_3.html#K8">`1</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="yellow_3.html#K8">`1</a> 
<b>by </b><i><a class="ref" href="yellow10.html#T14" target="_self">Th14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K4">Top</a> <span class="p2"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3">:]</a></span></span>)</span> <a href="yellow_3.html#K8">`1</a> 
<b>by </b><i><a class="txt" href="yellow10.html#E1:20_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="waybel_1.html#D23">WAYBEL_1:def 23</a></i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K8">`1</a> 
<b>by </b><i><a class="ref" href="yellow10.html#T3" target="_self">Th3</a></i>
<br/>.= 
 <a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D23">WAYBEL_1:def 23</a><br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="yellow_3.html#K8">`1</a> </span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="yellow_3.html#K8">`1</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="yellow_3.html#K8">`1</a> 
<b>by </b><i><a class="ref" href="yellow10.html#T13" target="_self">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <span class="p2"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3">:]</a></span></span>)</span> <a href="yellow_3.html#K8">`1</a> 
<b>by </b><i><a class="txt" href="yellow10.html#E1:20_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="waybel_1.html#D23">WAYBEL_1:def 23</a></i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K8">`1</a> 
<b>by </b><i><a class="ref" href="yellow10.html#T4" target="_self">Th4</a></i>
<br/>.= 
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:20_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="yellow_3.html#K9">`2</a>  <a href="yellow10.html#R1" target="_self">is_a_complement_of</a> <font color="Maroon">c<sub>4</sub></font> <a href="yellow_3.html#K9">`2</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2"><b>proof </b></a><div class="add">

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="yellow_3.html#K9">`2</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="yellow_3.html#K9">`2</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="yellow_3.html#K9">`2</a> 
<b>by </b><i><a class="ref" href="yellow10.html#T14" target="_self">Th14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K4">Top</a> <span class="p2"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3">:]</a></span></span>)</span> <a href="yellow_3.html#K9">`2</a> 
<b>by </b><i><a class="txt" href="yellow10.html#E1:20_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="waybel_1.html#D23">WAYBEL_1:def 23</a></i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K9">`2</a> 
<b>by </b><i><a class="ref" href="yellow10.html#T3" target="_self">Th3</a></i>
<br/>.= 
 <a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D23">WAYBEL_1:def 23</a><br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="yellow_3.html#K9">`2</a> </span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="yellow_3.html#K9">`2</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="yellow_3.html#K9">`2</a> 
<b>by </b><i><a class="ref" href="yellow10.html#T13" target="_self">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <span class="p2"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3">:]</a></span></span>)</span> <a href="yellow_3.html#K9">`2</a> 
<b>by </b><i><a class="txt" href="yellow10.html#E1:20_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="waybel_1.html#D23">WAYBEL_1:def 23</a></i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K9">`2</a> 
<b>by </b><i><a class="ref" href="yellow10.html#T4" target="_self">Th4</a></i>
<br/>.= 
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>


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

<a NAME="E2:20"/><i><font color="Green">E9</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <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="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E3:20"/><i><font color="Green">E10</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="yellow_3.html#K8">`1</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="yellow_3.html#K8">`1</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E4:20"/><i><font color="Green">E11</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="yellow_3.html#K8">`1</a> </span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="yellow_3.html#K8">`1</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E5:20"/><i><font color="Green">E12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="yellow_3.html#K9">`2</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="yellow_3.html#K9">`2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E6:20"/><i><font color="Green">E13</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="yellow_3.html#K9">`2</a> </span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="yellow_3.html#K9">`2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>2</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D23">WAYBEL_1:def 23</a><br/>

<i><font color="Green">E14</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="yellow_3.html#K8">`1</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="yellow_3.html#K8">`1</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="yellow_3.html#K8">`1</a> </span>)</span>
<b>by </b><i><a class="ref" href="yellow10.html#T14" target="_self">Th14</a></i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K8">`1</a> 
<b>by </b><i><a class="txt" href="yellow10.html#E3:20"><i><font color="Green">E10</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="yellow_3.html#K9">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="yellow_3.html#K9">`2</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="yellow_3.html#K9">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="yellow10.html#T14" target="_self">Th14</a></i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K9">`2</a> 
<b>by </b><i><a class="txt" href="yellow10.html#E5:20"><i><font color="Green">E12</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>
<b>hence </b><font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="yellow_3.html#K7">]</a></span>
<b>by </b><i><a class="txt" href="yellow10.html#E2:20"><i><font color="Green">E9</font></i></a>, <a class="txt" href="yellow10.html#E7:20"><i><font color="Green">E14</font></i></a>, <a class="ref" href="domain_1.html#T12">DOMAIN_1:12</a></i>
<br/>.= 
 <a href="yellow_0.html#K4">Top</a> <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3">:]</a></span>
<b>by </b><i><a class="ref" href="yellow10.html#T3" target="_self">Th3</a></i>
;<br/> <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D23">WAYBEL_1:def 23</a><br/>

<i><font color="Green">E15</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="yellow_3.html#K8">`1</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="yellow_3.html#K8">`1</a> </span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="yellow_3.html#K8">`1</a> </span>)</span>
<b>by </b><i><a class="ref" href="yellow10.html#T13" target="_self">Th13</a></i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K8">`1</a> 
<b>by </b><i><a class="txt" href="yellow10.html#E4:20"><i><font color="Green">E11</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="yellow_3.html#K9">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="yellow_3.html#K9">`2</a> </span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="yellow_3.html#K9">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="yellow10.html#T13" target="_self">Th13</a></i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K9">`2</a> 
<b>by </b><i><a class="txt" href="yellow10.html#E6:20"><i><font color="Green">E13</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>
<b>hence </b><font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="yellow_3.html#K7">]</a></span>
<b>by </b><i><a class="txt" href="yellow10.html#E2:20"><i><font color="Green">E9</font></i></a>, <a class="txt" href="yellow10.html#E10:20"><i><font color="Green">E15</font></i></a>, <a class="ref" href="domain_1.html#T12">DOMAIN_1:12</a></i>
<br/>.= 
 <a href="yellow_0.html#K3">Bottom</a> <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3">:]</a></span>
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