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<b>let </b><font color="Maroon">S</font> be   <a href="waybel_0.html#NM6">Semilattice</a>, <font color="Maroon">T</font> be   <a href="waybel_0.html#NM6">Semilattice</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:8"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">f</font> is <a href="waybel_0.html#V17">infs-preserving</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">e</font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="waybel21.html#D1" target="_self">WAYBEL21:def 1</a>
<div class="add"><b>assume </b><a NAME="E1:8_1"/>
<font color="Maroon">S</font> is <a href="yellow_0.html#V2">upper-bounded</a>
 ;<br/><a NAME="E2:8_1"/><b>then </b>
(  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">e</font>,<font color="Maroon">S</font> &amp; <font color="Maroon">f</font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Maroon">e</font> )
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D32">WAYBEL_0:def 32</a>, <a class="ref" href="yellow_0.html#T43">YELLOW_0:43</a></i>;<br/><a NAME="E3:8_1"/><b>then </b>
(  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">e</font>,<font color="Maroon">T</font> &amp; <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="relat_1.html#T149">RELAT_1:149</a>, <a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a></i>;<br/><b>hence </b><a NAME="E4:8_1"/>
<font color="Maroon">T</font> is <a href="yellow_0.html#V2">upper-bounded</a>
 <b>by </b><i><a class="ref" href="waybel20.html#T5">WAYBEL20:5</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">X</font> be  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/>

<b>thus </b><a NAME="E4:8"/>
<font color="Maroon">f</font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D32">WAYBEL_0:def 32</a></i>;<br/>


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