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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:23_1_1"/><i><font color="Green" title="E14">A1</font></i>: 
for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   ex <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b2">a</font> &amp; ( for <font color="Olive" title="b3">b</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">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b3">b</font> holds <br/><font color="Olive" title="b2">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">b</font> ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D12" title="LATTICE3:def.12">LATTICE3:def 12</a><br/>

<b>then consider </b><font color="Maroon" title="c2">a0</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="E3:23_1_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c2">a0</font>
 <b>and </b><br/><a NAME="E4:23_1_1"/><i><font color="Green" title="E15">A2</font></i>: 
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c2">a0</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">b</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c3">a1</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="E6:23_1_1"/><i><font color="Green" title="E16">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c3">a1</font>
 <b>and </b><br/><a NAME="E7:23_1_1"/>
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c3">a1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">b</font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E1:23_1_1"><i><font color="Green" title="E14">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">a0</font> = <font color="Maroon" title="c2">a0</font>, <font color="Maroon" title="c5">a1</font> = <font color="Maroon" title="c3">a1</font> as   <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> <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D4" target="_self" title="YELLOW_0:def.4">YELLOW_0:def 4</a>,<a class="ref" href="yellow_0.html#D6" target="_self" title="YELLOW_0:def.6">YELLOW_0:def 6</a>
<div class="add"><b>take </b><font color="Maroon" title="c6">a0</font> = <font color="Maroon" title="c4">a0</font>;<br/><b>thus </b><a NAME="E1:23_1_1_1"/>
<font color="Maroon" title="c6">a0</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E1:23_1_1_1_1"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c7">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E3:23_1_1_1_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c8">b</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T6" target="_self" title="YELLOW_0:th.6">Th6</a></i>;<br/>
<b>hence </b><a NAME="E4:23_1_1_1_1"/>
<font color="Maroon" title="c6">a0</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E4:23_1_1"><i><font color="Green" title="E15">A2</font></i></a></i>;<br/>


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

<b>take </b>
<font color="Maroon" title="c5">a1</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D5" target="_self" title="YELLOW_0:def.5">YELLOW_0:def 5</a><br/>

<b>thus </b><a NAME="E10:23_1_1"/>
<font color="Maroon" title="c5">a1</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E6:23_1_1"><i><font color="Green" title="E16">A3</font></i></a></i>;<br/>


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