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<b>let </b><font color="Maroon" title="c1">CL</font> be   <a href="lattices.html#NM7" title="LATTICES:NM.7">C_Lattice</a>;<br/><b>let </b><font color="Maroon" title="c2">BL</font> be  <a href="lattices.html#V17" title="LATTICES:attr.17">Boolean</a> <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be    <a href="lattice4.html#M1" title="LATTICE4:mode.1">Homomorphism</a> of <font color="Maroon" title="c2">BL</font>,<font color="Maroon" title="c1">CL</font>;<br/>





<b>assume </b><a NAME="E1:41"/><i><font color="Green" title="E35">A1</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="lattice4.html#V2" title="LATTICE4:attr.2">epimorphism</a>
 ;<br/>

<a NAME="E2:41"/><b>then </b><i><font color="Green" title="E36">A2</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="lattice4.html#R2" title="LATTICE4:pred.2">preserves_top</a> 
 
<b>by </b><i><a class="ref" href="lattice4.html#T25" target="_self" title="LATTICE4:th.25">Th25</a></i>;<br/>
<a NAME="E3:41"/><i><font color="Green" title="E37">A3</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="lattice4.html#R3" title="LATTICE4:pred.3">preserves_bottom</a> 
 
<b>by </b><i><a class="txt" href="lattice4.html#E1:41"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="lattice4.html#T19" target="_self" title="LATTICE4:th.19">Th19</a></i>;<br/>
<b>thus </b><a NAME="E4:41"/>
( <font color="Maroon" title="c1">CL</font> is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a> &amp; <font color="Maroon" title="c1">CL</font> is <a href="lattices.html#V16" title="LATTICES:attr.16">complemented</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D20" title="LATTICES:def.20">LATTICES:def 20</a><br/>

<b>thus </b><a NAME="E5:41"/>
<font color="Maroon" title="c1">CL</font> is <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a>
 <b>by </b><i><a class="txt" href="lattice4.html#E1:41"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="lattice4.html#T18" target="_self" title="LATTICE4:th.18">Th18</a></i>;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">CL</font> = <font color="Maroon" title="c1">CL</font> as  <a href="lattices.html#V17" title="LATTICES:attr.17">Boolean</a> <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a> <b>by </b><i><a class="ref" href="lattices.html#D20" title="LATTICES:def.20">LATTICES:def 20</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c3">f</font> as    <a href="lattice4.html#M1" title="LATTICE4:mode.1">Homomorphism</a> of <font color="Maroon" title="c2">BL</font>,<font color="Maroon" title="c4">CL</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">a1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">BL</font>;<br/><i><font color="Green" title="E38">A4</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a1</font></span>)</span> = 
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">a1</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice4.html#D1" target="_self" title="LATTICE4:def.1">Def1</a></i>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c2">BL</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#T48" title="LATTICES:th.48">LATTICES:48</a></i>
<br/>.= 
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c4">CL</font>
<b>by </b><i><a class="txt" href="lattice4.html#E2:41"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="lattice4.html#D7" target="_self" title="LATTICE4:def.7">Def7</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a1</font></span>)</span> = 
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">a1</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice4.html#D1" target="_self" title="LATTICE4:def.1">Def1</a></i>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c2">BL</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#T47" title="LATTICES:th.47">LATTICES:47</a></i>
<br/>.= 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c4">CL</font>
<b>by </b><i><a class="txt" href="lattice4.html#E3:41"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="lattice4.html#D8" target="_self" title="LATTICE4:def.8">Def8</a></i>
;<br/><a NAME="E3:41_1"/><b>then </b>
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a1</font>
 <b>by </b><i><a class="txt" href="lattice4.html#E1:41_1"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</a></i>;<br/><b>hence </b><a NAME="E4:41_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a1</font></span>)</span> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="lattices.html#D21" title="LATTICES:def.21">LATTICES:def 21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:41"/>
<font color="Maroon" title="c3">f</font> <a href="lattice4.html#R4" title="LATTICE4:pred.4">preserves_complement</a> 
 <b>by </b><i><a class="ref" href="lattice4.html#D9" target="_self" title="LATTICE4:def.9">Def9</a></i>;<br/>


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