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<b>let </b><font color="Maroon">L</font> be  <a href="orders_2.html#V3">transitive</a> <a href="orders_2.html#V4">antisymmetric</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:49"/><i><font color="Green">E24</font></i>: 
(  <a href="yellow_0.html#R1" target="_self">ex_sup_of</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#R1" target="_self">ex_sup_of</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#R1" target="_self">ex_sup_of</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font> )
 ;<br/>

<b>set </b><font color="Maroon">a</font> =  <a href="yellow_0.html#K1" target="_self">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>;<br/>
<b>set </b><font color="Maroon">b</font> =  <a href="yellow_0.html#K1" target="_self">"\/"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font>;<br/>
<b>set </b><font color="Maroon">c</font> =  <a href="yellow_0.html#K1" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">L</font>;<br/>
<a NAME="E2:49"/>
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E3:49"/><b>then </b><i><font color="Green">E29</font></i>: 
(  <a href="yellow_0.html#K1" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">L</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#K1" target="_self">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#K1" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">L</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#K1" target="_self">"\/"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font> )
 
<b>by </b><i>, <a class="txt" href="yellow_0.html#E4:49"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E4:49"/><i><font color="Green">E30</font></i>: 
(  <a href="yellow_0.html#K1" target="_self">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font> &amp;  <a href="yellow_0.html#K1" target="_self">"\/"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">Y</font> )
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>now </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">L</font>;<br/><b>assume </b><a NAME="E1:49_1"/>
( <font color="Maroon">d</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#K1" target="_self">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> &amp; <font color="Maroon">d</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#K1" target="_self">"\/"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font> )
 ;<br/><a NAME="E2:49_1"/><b>then </b>
( <font color="Maroon">d</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font> &amp; <font color="Maroon">d</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E3:49_1"/><b>then </b>
<font color="Maroon">d</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:49_1"/>
 <a href="yellow_0.html#K1" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">L</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">d</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:49"/>
 <a href="yellow_0.html#K1" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1" target="_self">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K10">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1" target="_self">"\/"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


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