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<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/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>





<a NAME="E1:20"/>
(  <a href="lfuzzy_0.html#K3" target="_self">min</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="lfuzzy_0.html#NR5" target="_self">&lt;R=</a> <font color="Maroon">a</font> &amp;  <a href="lfuzzy_0.html#K3" target="_self">min</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="lfuzzy_0.html#NR5" target="_self">&lt;R=</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/>
<a NAME="E2:20"/><b>then </b><i><font color="Green">E41</font></i>: 
(  <a href="lfuzzy_0.html#K3" target="_self">min</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">a</font> &amp;  <a href="lfuzzy_0.html#K3" target="_self">min</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">b</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:20"/>
for <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>  st <font color="Olive">d</font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Olive">d</font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">b</font> holds <br/> <a href="lfuzzy_0.html#K3" target="_self">min</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="lfuzzy_0.html#NR2" target="_self">&gt;&gt;=</a> <font color="Olive">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>

<b>assume </b><a NAME="E1:20_1"/>
( <font color="Maroon">d</font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">d</font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">b</font> )
 ;<br/>

<a NAME="E2:20_1"/><b>then </b>
( <font color="Maroon">d</font> <a href="lfuzzy_0.html#NR5" target="_self">&lt;R=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">d</font> <a href="lfuzzy_0.html#NR5" target="_self">&lt;R=</a> <font color="Maroon">b</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:20_1"/><b>then </b>
<font color="Maroon">d</font> <a href="lfuzzy_0.html#NR5" target="_self">&lt;R=</a>  <a href="lfuzzy_0.html#K3" target="_self">min</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="square_1.html#T39">SQUARE_1:39</a></i>;<br/>
<b>hence </b><a NAME="E4:20_1"/>
 <a href="lfuzzy_0.html#K3" target="_self">min</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="lfuzzy_0.html#NR2" target="_self">&gt;&gt;=</a> <font color="Maroon">d</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:20"/>
<font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="lfuzzy_0.html#K3" target="_self">min</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>


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