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

<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a> <a href="lfuzzy_0.html#V1" target="_self">real</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>given </b><font color="Maroon">x</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><a NAME="E1:22_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; ( for <font color="Olive">y</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> holds <br/><font color="Maroon">x</font> <a href="lfuzzy_0.html#NR7" target="_self">&gt;R=</a> <font color="Olive">y</font> ) )
 ;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/><b>assume </b><a NAME="E1:22_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 ;<br/><a NAME="E2:22_1_1"/>
<font color="Maroon">a</font> <a href="lfuzzy_0.html#NR7" target="_self">&gt;R=</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:22_1_1"/>
<font color="Maroon">a</font> <a href="lfuzzy_0.html#NR2" target="_self">&gt;&gt;=</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E4:22_1"/><b>then </b>
<font color="Maroon">a</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/><b>hence </b><a NAME="E5:22_1"/>
<font color="Maroon">R</font> is <a href="yellow_0.html#V2">upper-bounded</a>
 <b>by </b><i><a class="ref" href="yellow_0.html#D5">YELLOW_0:def 5</a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><a NAME="E2:22"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D5">YELLOW_0:def 5</a><br/>

<b>take </b>
<font color="Maroon">x</font>
;<br/>

<b>thus </b><a NAME="E3:22"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 ;<br/>

<b>let </b><font color="Maroon">y</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E4:22"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> ;<br/>
<a NAME="E6:22"/>
<font color="Maroon">x</font> <a href="lfuzzy_0.html#NR2" target="_self">&gt;&gt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E7:22"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/>


</div>
