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<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V4">antisymmetric</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:7"/><i><font color="Green">E30</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">S</font>
 ;<br/>

<b>take </b>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">S</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D4">YELLOW_0:def 4</a><br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a><br/>

<a NAME="E2:7"/>
(  <a href="xboole_0.html#K1">{}</a>  <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">x</font> &amp;  <a href="yellow_0.html#K1">"\/"</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">S</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T6">YELLOW_0:6</a></i>;<br/>
<b>hence </b><a NAME="E3:7"/>
(  not <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">S</font> or  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">S</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/>


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