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

<b>let </b><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</font> be  <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a> <a href="yellow_0.html#V3" title="YELLOW_0:attr.3">bounded</a> <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a> <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="yellow10.html#R1" title="YELLOW10:pred.1">is_a_complement_of</a> <font color="Olive" title="b2">y</font> iff ( <font color="Olive" title="b1">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  <a href="yellow10.html#R1" title="YELLOW10:pred.1">is_a_complement_of</a> <font color="Olive" title="b2">y</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  &amp; <font color="Olive" title="b1">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  <a href="yellow10.html#R1" title="YELLOW10:pred.1">is_a_complement_of</a> <font color="Olive" title="b2">y</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  ) )</span><br/><b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="yellow10.html#R1" title="YELLOW10:pred.1">is_a_complement_of</a> <font color="Maroon" title="c4">y</font> iff ( <font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  <a href="yellow10.html#R1" title="YELLOW10:pred.1">is_a_complement_of</a> <font color="Maroon" title="c4">y</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  &amp; <font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  <a href="yellow10.html#R1" title="YELLOW10:pred.1">is_a_complement_of</a> <font color="Maroon" title="c4">y</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  ) )</span><br/>



<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  <a href="yellow10.html#R1" title="YELLOW10:pred.1">is_a_complement_of</a> <font color="Maroon" title="c4">y</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  &amp; <font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  <a href="yellow10.html#R1" title="YELLOW10:pred.1">is_a_complement_of</a> <font color="Maroon" title="c4">y</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  implies <font color="Maroon" title="c3">x</font> <a href="yellow10.html#R1" title="YELLOW10:pred.1">is_a_complement_of</a> <font color="Maroon" title="c4">y</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:20_1"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="yellow10.html#R1" title="YELLOW10:pred.1">is_a_complement_of</a> <font color="Maroon" title="c4">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  <a href="yellow10.html#R1" title="YELLOW10:pred.1">is_a_complement_of</a> <font color="Maroon" title="c4">y</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  &amp; <font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  <a href="yellow10.html#R1" title="YELLOW10:pred.1">is_a_complement_of</a> <font color="Maroon" title="c4">y</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  )</span><br/><b>thus </b><a NAME="E2:20_1"/>
<font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  <a href="yellow10.html#R1" title="YELLOW10:pred.1">is_a_complement_of</a> <font color="Maroon" title="c4">y</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  <a href="yellow10.html#R1" title="YELLOW10:pred.1">is_a_complement_of</a> <font color="Maroon" title="c4">y</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span><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" title="c4">y</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> </span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> 
<b>by </b><i><a class="ref" href="yellow10.html#T14" target="_self" title="YELLOW10:th.14">Th14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <span class="p2"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span></span>)</span> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> 
<b>by </b><i><a class="txt" href="#E1:20_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="waybel_1.html#D23" title="WAYBEL_1:def.23">WAYBEL_1:def 23</a></i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c2">T</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> 
<b>by </b><i><a class="ref" href="yellow10.html#T3" target="_self" title="YELLOW10:th.3">Th3</a></i>
<br/>.= 
 <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">S</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D23" title="WAYBEL_1:def.23">WAYBEL_1:def 23</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> </span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">S</font></span><br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> </span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> 
<b>by </b><i><a class="ref" href="yellow10.html#T13" target="_self" title="YELLOW10:th.13">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p2"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span></span>)</span> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> 
<b>by </b><i><a class="txt" href="#E1:20_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="waybel_1.html#D23" title="WAYBEL_1:def.23">WAYBEL_1:def 23</a></i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">T</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> 
<b>by </b><i><a class="ref" href="yellow10.html#T4" target="_self" title="YELLOW10:th.4">Th4</a></i>
<br/>.= 
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">S</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:20_1"/>
<font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  <a href="yellow10.html#R1" title="YELLOW10:pred.1">is_a_complement_of</a> <font color="Maroon" title="c4">y</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="c4">y</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> 
<b>by </b><i><a class="ref" href="yellow10.html#T14" target="_self" title="YELLOW10:th.14">Th14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <span class="p2"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span></span>)</span> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> 
<b>by </b><i><a class="txt" href="#E1:20_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="waybel_1.html#D23" title="WAYBEL_1:def.23">WAYBEL_1:def 23</a></i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c2">T</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> 
<b>by </b><i><a class="ref" href="yellow10.html#T3" target="_self" title="YELLOW10:th.3">Th3</a></i>
<br/>.= 
 <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c2">T</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D23" title="WAYBEL_1:def.23">WAYBEL_1:def 23</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">T</font></span><br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> 
<b>by </b><i><a class="ref" href="yellow10.html#T13" target="_self" title="YELLOW10:th.13">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p2"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span></span>)</span> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> 
<b>by </b><i><a class="txt" href="#E1:20_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="waybel_1.html#D23" title="WAYBEL_1:def.23">WAYBEL_1:def 23</a></i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">T</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> 
<b>by </b><i><a class="ref" href="yellow10.html#T4" target="_self" title="YELLOW10:th.4">Th4</a></i>
<br/>.= 
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">T</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<a NAME="E2:20"/><i><font color="Green" title="E9">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#D2" title="YELLOW_3:def.2">YELLOW_3:def 2</a></i>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E3:20"/><i><font color="Green" title="E10">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> </span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">S</font>
 <b>and </b><br/><a NAME="E4:20"/><i><font color="Green" title="E11">A4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> </span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">S</font>
 <b>and </b><br/><a NAME="E5:20"/><i><font color="Green" title="E12">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c2">T</font>
 <b>and </b><br/><a NAME="E6:20"/><i><font color="Green" title="E13">A6</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">T</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D23" title="WAYBEL_1:def.23">WAYBEL_1:def 23</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="yellow10.html#R1" title="YELLOW10:pred.1">is_a_complement_of</a> <font color="Maroon" title="c4">y</font></span><br/>

<i><font color="Green" title="E14">A7</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> </span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> </span>)</span>
<b>by </b><i><a class="ref" href="yellow10.html#T14" target="_self" title="YELLOW10:th.14">Th14</a></i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c2">T</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> 
<b>by </b><i><a class="txt" href="#E3:20"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="yellow10.html#T14" target="_self" title="YELLOW10:th.14">Th14</a></i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c2">T</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> 
<b>by </b><i><a class="txt" href="#E5:20"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/>
<b>hence </b><font color="Maroon" title="c4">y</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c3">x</font> = 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c2">T</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
<b>by </b><i><a class="txt" href="#E2:20"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="#E7:20"><i><font color="Green" title="E14">A7</font></i></a>, <a class="ref" href="domain_1.html#T12" title="DOMAIN_1:th.12">DOMAIN_1:12</a></i>
<br/>.= 
 <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>
<b>by </b><i><a class="ref" href="yellow10.html#T3" target="_self" title="YELLOW10:th.3">Th3</a></i>
;<br/> <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D23" title="WAYBEL_1:def.23">WAYBEL_1:def 23</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">y</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span></span><br/>

<i><font color="Green" title="E15">A8</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> </span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> </span>)</span>
<b>by </b><i><a class="ref" href="yellow10.html#T13" target="_self" title="YELLOW10:th.13">Th13</a></i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">T</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> 
<b>by </b><i><a class="txt" href="#E4:20"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="yellow10.html#T13" target="_self" title="YELLOW10:th.13">Th13</a></i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">T</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> 
<b>by </b><i><a class="txt" href="#E6:20"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/>
<b>hence </b><font color="Maroon" title="c4">y</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c3">x</font> = 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">T</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
<b>by </b><i><a class="txt" href="#E2:20"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="#E10:20"><i><font color="Green" title="E15">A8</font></i></a>, <a class="ref" href="domain_1.html#T12" title="DOMAIN_1:th.12">DOMAIN_1:12</a></i>
<br/>.= 
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>
<b>by </b><i><a class="ref" href="yellow10.html#T4" target="_self" title="YELLOW10:th.4">Th4</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
