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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:31_1"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c3">b</font>
 <b>and </b><br/><a NAME="E2:31_1"/><i><font color="Green" title="E23">A2</font></i>: 
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c1">L</font>
 ;<br/><b>consider </b><font color="Maroon" title="c5">c1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E4:31_1"/><i><font color="Green" title="E24">A3</font></i>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c5">c1</font>
 <b>and </b><br/><a NAME="E5:31_1"/><i><font color="Green" title="E25">A4</font></i>: 
for <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Olive" title="b1">d</font> holds <br/><font color="Maroon" title="c5">c1</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">d</font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E2:31_1"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="yellow_0.html#T16" target="_self" title="YELLOW_0:th.16">Th16</a></i>;<br/><div><i><font color="Green" title="E26">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/><b>assume </b><a NAME="E1:31_1_1"/>
( <font color="Maroon" title="c2">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">d</font> &amp; <font color="Maroon" title="c3">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">d</font> )
 ;<br/><a NAME="E2:31_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c6">d</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T8" target="_self" title="YELLOW_0:th.8">Th8</a></i>;<br/><b>hence </b><a NAME="E3:31_1_1"/>
<font color="Maroon" title="c5">c1</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">d</font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E5:31_1"><i><font color="Green" title="E25">A4</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E7:31_1"/>
( <font color="Maroon" title="c2">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">c1</font> &amp; <font color="Maroon" title="c3">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">c1</font> )
 <b>by </b><i><a class="txt" href="yellow_0.html#E4:31_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="yellow_0.html#T8" target="_self" title="YELLOW_0:th.8">Th8</a></i>;<br/><b>hence </b><a NAME="E8:31_1"/>
( <font color="Maroon" title="c4">c</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">c</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> &amp; ( for <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">d</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b1">d</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> holds <br/><font color="Maroon" title="c4">c</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">d</font> ) )
 <b>by </b><i><a class="txt" href="yellow_0.html#E1:31_1"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="yellow_0.html#E6:31_1"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="lattice3.html#D14" title="LATTICE3:def.14">LATTICE3:def 14</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:31"/><i><font color="Green" title="E22">A6</font></i>: 
( <font color="Maroon" title="c4">c</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">c</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> &amp; ( for <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">d</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b1">d</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> holds <br/><font color="Maroon" title="c4">c</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">d</font> ) )
 ;<br/>

<b>hence </b><a NAME="E3:31"/>
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D14" title="LATTICE3:def.14">LATTICE3:def 14</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c5">c</font> = <font color="Maroon" title="c4">c</font>;<br/><b>thus </b><a NAME="E1:31_2"/>
<font color="Maroon" title="c5">c</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="yellow_0.html#E2:31"><i><font color="Green" title="E22">A6</font></i></a>, <a class="ref" href="yellow_0.html#T8" target="_self" title="YELLOW_0:th.8">Th8</a></i>;<br/><b>let </b><font color="Maroon" title="c6">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/><b>assume </b><a NAME="E2:31_2"/>
<font color="Maroon" title="c6">d</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ;<br/><a NAME="E3:31_2"/><b>then </b>
( <font color="Maroon" title="c6">d</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c6">d</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T8" target="_self" title="YELLOW_0:th.8">Th8</a></i>;<br/><b>hence </b><a NAME="E4:31_2"/>
<font color="Maroon" title="c5">c</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">d</font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E2:31"><i><font color="Green" title="E22">A6</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:31"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T16" target="_self" title="YELLOW_0:th.16">Th16</a></i>;<br/>


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