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<b>let </b><font color="Maroon">L2</font> be   <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">DL</font> be  <a href="lattices.html#V11">distributive</a> <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="lattice4.html#M1" target="_self">Homomorphism</a> of <font color="Maroon">DL</font>,<font color="Maroon">L2</font>;<br/>





<b>assume </b><a NAME="E1:20"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">f</font> is <a href="lattice4.html#V2" target="_self">epimorphism</a>
 ;<br/>

<b>thus </b><a NAME="E2:20"/>
<font color="Maroon">L2</font> is <a href="lattices.html#V11">distributive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font>, <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D11">LATTICES:def 11</a><br/>

<b>consider </b><font color="Maroon">a'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">DL</font><b> such that </b><br/><a NAME="E2:20_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">b'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">DL</font><b> such that </b><br/><a NAME="E4:20_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">c'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">DL</font><b> such that </b><br/><a NAME="E6:20_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i>, </i>;<br/>
<b>thus </b><font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c</font></span>)</span> = 
<font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b'</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c'</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">b'</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c'</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b'</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D11">LATTICES:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">a'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b'</font></span>)</span></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">a'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c'</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">a'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c'</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i>, , </i>
;<br/>


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


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