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<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>, <font color="Maroon">M</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<font color="Maroon">M</font>;<br/>



<b>assume </b><a NAME="E1:41"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">f</font> is <a href="waybel_0.html#V18">sups-preserving</a> or <font color="Maroon">f</font> is <a href="waybel_0.html#V17">infs-preserving</a> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1"/>
( <font color="Maroon">f</font> is <a href="waybel_0.html#V18">sups-preserving</a> or <font color="Maroon">f</font> is <a href="waybel_0.html#V17">infs-preserving</a> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1"><b>suppose </b></a><a NAME="E1:41_1_1"/>
<font color="Maroon">f</font> is <a href="waybel_0.html#V18">sups-preserving</a>
 ;<br/><div class="add"><a NAME="E2:41_1_1"/><b>then </b>
 <a href="yellow_2.html#K1" target="_self">Image</a> <font color="Maroon">f</font> is <a href="yellow_0.html#V8">sups-inheriting</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:41_1_1"/>
 <a href="yellow_2.html#K1" target="_self">Image</a> <font color="Maroon">f</font> is  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2"><b>suppose </b></a><a NAME="E1:41_1_2"/>
<font color="Maroon">f</font> is <a href="waybel_0.html#V17">infs-preserving</a>
 ;<br/><div class="add"><a NAME="E2:41_1_2"/><b>then </b>
 <a href="yellow_2.html#K1" target="_self">Image</a> <font color="Maroon">f</font> is <a href="yellow_0.html#V7">infs-inheriting</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:41_1_2"/>
 <a href="yellow_2.html#K1" target="_self">Image</a> <font color="Maroon">f</font> is  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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