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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="lattices.html#V11">distributive</a> <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="lattice4.html#M1" target="_self">Homomorphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:20"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="lattice4.html#V2" target="_self">epimorphism</a>
 ;<br/>

<b>thus </b><a NAME="E2:20"/>
<font color="Maroon">c<sub>1</sub></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">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E2:20_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="lattice4.html#E1:20"><i><font color="Green">E19</font></i></a>, <a class="ref" href="lattice4.html#T9" target="_self">Th9</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:20_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="lattice4.html#E1:20"><i><font color="Green">E19</font></i></a>, <a class="ref" href="lattice4.html#T9" target="_self">Th9</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:20_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="lattice4.html#E1:20"><i><font color="Green">E19</font></i></a>, <a class="ref" href="lattice4.html#T9" target="_self">Th9</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
<font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lattice4.html#E4:20_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="lattice4.html#E6:20_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="lattice4.html#D1" target="_self">Def1</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lattice4.html#E2:20_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="lattice4.html#D1" target="_self">Def1</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>9</sub></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">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice4.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lattice4.html#E2:20_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="lattice4.html#E4:20_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="lattice4.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="lattice4.html#E2:20_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="lattice4.html#E6:20_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="lattice4.html#D1" target="_self">Def1</a></i>
;<br/>


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


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