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

<b>let </b><font color="Maroon" title="c1">L1</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a> <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">L2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>;<br/><b>let </b><font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font>;<br/>







<b>assume </b><a NAME="E1:16"/><i><font color="Green" title="E11">A1</font></i>: 
(  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">L2</font> #) &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> )
 ;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>
<div class="add"><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:16_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">x</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font><b> such that </b><br/><a NAME="E3:16_1_1"/><i><font color="Green" title="E12">A2</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z</font> &amp; <font color="Maroon" title="c6">z</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c3">x</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c7">w</font> = <font color="Maroon" title="c6">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font> <b>by </b><i><a class="txt" href="yellow12.html#E1:16"><i><font color="Green" title="E11">A1</font></i></a></i>;<br/><a NAME="E5:16_1_1"/>
<font color="Maroon" title="c2">L2</font> is <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>
 <b>by </b><i><a class="txt" href="yellow12.html#E1:16"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="waybel_8.html#T14" title="WAYBEL_8:th.14">WAYBEL_8:14</a></i>;<br/><a NAME="E6:16_1_1"/><b>then </b>
<font color="Maroon" title="c7">w</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="yellow12.html#E1:16"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="yellow12.html#E3:16_1_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="waybel_8.html#T8" title="WAYBEL_8:th.8">WAYBEL_8:8</a></i>;<br/><b>hence </b><a NAME="E7:16_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="yellow12.html#E3:16_1_1"><i><font color="Green" title="E12">A2</font></i></a></i>;<br/></div>
<b>end;</b></div><b>let </b><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E2:16_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c4">y</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font><b> such that </b><br/><a NAME="E4:16_1"/><i><font color="Green" title="E12">A3</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z</font> &amp; <font color="Maroon" title="c6">z</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c4">y</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c7">w</font> = <font color="Maroon" title="c6">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> <b>by </b><i><a class="txt" href="yellow12.html#E1:16"><i><font color="Green" title="E11">A1</font></i></a></i>;<br/><a NAME="E6:16_1"/>
( <font color="Maroon" title="c2">L2</font> is <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> &amp; <font color="Maroon" title="c2">L2</font> is <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a> )
 <b>by </b><i><a class="txt" href="yellow12.html#E1:16"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="waybel_8.html#T14" title="WAYBEL_8:th.14">WAYBEL_8:14</a>, <a class="ref" href="waybel_8.html#T15" title="WAYBEL_8:th.15">WAYBEL_8:15</a></i>;<br/><a NAME="E7:16_1"/><b>then </b>
<font color="Maroon" title="c7">w</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="yellow12.html#E1:16"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="yellow12.html#E4:16_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="waybel_8.html#T8" title="WAYBEL_8:th.8">WAYBEL_8:8</a></i>;<br/><b>hence </b><a NAME="E8:16_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="yellow12.html#E4:16_1"><i><font color="Green" title="E12">A3</font></i></a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:16_2"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c3">x</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font><b> such that </b><br/><a NAME="E3:16_2"/><i><font color="Green" title="E12">A4</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z</font> &amp; <font color="Maroon" title="c6">z</font> <a href="waybel_3.html#NR3" title="WAYBEL_3:NR.3">&gt;&gt;</a> <font color="Maroon" title="c3">x</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c7">w</font> = <font color="Maroon" title="c6">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font> <b>by </b><i><a class="txt" href="yellow12.html#E1:16"><i><font color="Green" title="E11">A1</font></i></a></i>;<br/><a NAME="E5:16_2"/>
<font color="Maroon" title="c2">L2</font> is <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>
 <b>by </b><i><a class="txt" href="yellow12.html#E1:16"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="waybel_8.html#T14" title="WAYBEL_8:th.14">WAYBEL_8:14</a></i>;<br/><a NAME="E6:16_2"/><b>then </b>
<font color="Maroon" title="c7">w</font> <a href="waybel_3.html#NR3" title="WAYBEL_3:NR.3">&gt;&gt;</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="yellow12.html#E1:16"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="yellow12.html#E3:16_2"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="waybel_8.html#T8" title="WAYBEL_8:th.8">WAYBEL_8:8</a></i>;<br/><b>hence </b><a NAME="E7:16_2"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="yellow12.html#E3:16_2"><i><font color="Green" title="E12">A4</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:16"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c4">y</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font><b> such that </b><br/><a NAME="E6:16"/><i><font color="Green" title="E12">A5</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z</font> &amp; <font color="Maroon" title="c6">z</font> <a href="waybel_3.html#NR3" title="WAYBEL_3:NR.3">&gt;&gt;</a> <font color="Maroon" title="c4">y</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">w</font> = <font color="Maroon" title="c6">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> <b>by </b><i><a class="txt" href="yellow12.html#E1:16"><i><font color="Green" title="E11">A1</font></i></a></i>;<br/>
<a NAME="E8:16"/>
( <font color="Maroon" title="c2">L2</font> is <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> &amp; <font color="Maroon" title="c2">L2</font> is <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a> )
 
<b>by </b><i><a class="txt" href="yellow12.html#E1:16"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="waybel_8.html#T14" title="WAYBEL_8:th.14">WAYBEL_8:14</a>, <a class="ref" href="waybel_8.html#T15" title="WAYBEL_8:th.15">WAYBEL_8:15</a></i>;<br/>
<a NAME="E9:16"/><b>then </b>
<font color="Maroon" title="c7">w</font> <a href="waybel_3.html#NR3" title="WAYBEL_3:NR.3">&gt;&gt;</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="txt" href="yellow12.html#E1:16"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="yellow12.html#E6:16"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="waybel_8.html#T8" title="WAYBEL_8:th.8">WAYBEL_8:8</a></i>;<br/>
<b>hence </b><a NAME="E10:16"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="yellow12.html#E6:16"><i><font color="Green" title="E12">A5</font></i></a></i>;<br/>


</div>
