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<b>let </b><font color="Maroon" title="c1">L2</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>;<br/><b>let </b><font color="Maroon" title="c2">1L</font> be  <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</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">1L</font>,<font color="Maroon" title="c1">L2</font>;<br/>





<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E26">A1</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="lattice4.html#V2" title="LATTICE4:attr.2">epimorphism</a>
 ;<br/>

<b>set </b><font color="Maroon" title="c4">r</font> = <font color="Maroon" title="c3">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">1L</font></span>)</span>;<br/>
<div><i><font color="Green" title="E27">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">a2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L2</font>;<br/><b>consider </b><font color="Maroon" title="c6">a1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">1L</font><b> such that </b><br/><a NAME="E2:27_1"/><i><font color="Green" title="E28">A3</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a2</font>
 <b>by </b><i><a class="txt" href="lattice4.html#E1:27"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="lattice4.html#T9" target="_self" title="LATTICE4:th.9">Th9</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c2">1L</font></span>)</span></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">a2</font> = 
<font color="Maroon" title="c3">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"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c2">1L</font></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="txt" href="lattice4.html#E2:27_1"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="lattice4.html#D1" target="_self" title="LATTICE4:def.1">Def1</a></i>
<br/>.= 
<font color="Maroon" title="c3">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">1L</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#T44" title="LATTICES:th.44">LATTICES:44</a></i>
;<br/><b>hence </b><a NAME="E4:27_1"/>
<font color="Maroon" title="c5">a2</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c2">1L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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">1L</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E3:27"/>
<font color="Maroon" title="c1">L2</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c3">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">1L</font></span>)</span>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D14" title="LATTICES:def.14">LATTICES:def 14</a><br/>

<b>thus </b><a NAME="E1:27_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L2</font> holds <br/> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c2">1L</font></span>)</span></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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">1L</font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c2">1L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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">1L</font></span>)</span> )
 <b>by </b><i><a class="txt" href="lattice4.html#E2:27"><i><font color="Green" title="E27">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>

<a NAME="E4:27"/><b>then </b>
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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">1L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice4.html#E2:27"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="lattices.html#D17" title="LATTICES:def.17">LATTICES:def 17</a></i>;<br/>
<b>hence </b><a NAME="E5:27"/>
<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#D7" target="_self" title="LATTICE4:def.7">Def7</a></i>;<br/>


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