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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">DL</font> being <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a> <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a><br/> for <font color="Olive" title="b2">f</font> being   <a href="lattice4.html#M1" title="LATTICE4:mode.1">Homomorphism</a> of <font color="Olive" title="b1">DL</font>,<font color="Maroon" title="c1">L2</font>  st <font color="Olive" title="b2">f</font> is <a href="lattice4.html#V2" title="LATTICE4:attr.2">epimorphism</a> holds <br/><font color="Maroon" title="c1">L2</font> is <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a></span><br/><b>let </b><font color="Maroon" title="c2">DL</font> be  <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a> <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being   <a href="lattice4.html#M1" title="LATTICE4:mode.1">Homomorphism</a> of <font color="Maroon" title="c2">DL</font>,<font color="Maroon" title="c1">L2</font>  st <font color="Olive" title="b1">f</font> is <a href="lattice4.html#V2" title="LATTICE4:attr.2">epimorphism</a> holds <br/><font color="Maroon" title="c1">L2</font> is <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a></span><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">DL</font>,<font color="Maroon" title="c1">L2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is <a href="lattice4.html#V2" title="LATTICE4:attr.2">epimorphism</a> implies <font color="Maroon" title="c1">L2</font> is <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a> )</span><br/>





<b>assume </b><a NAME="E1:20"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="lattice4.html#V2" title="LATTICE4:attr.2">epimorphism</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">L2</font> is <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a></span><br/>

<b>thus </b><a NAME="E2:20"/>
<font color="Maroon" title="c1">L2</font> is <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D11" title="LATTICES:def.11">LATTICES:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c6">c</font></span>)</span></span><br/>

<b>consider </b><font color="Maroon" title="c7">a'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">DL</font><b> such that </b><br/><a NAME="E2:20_1"/><i><font color="Green" title="E18">A2</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="c7">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="lattice4.html#E1:20"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="lattice4.html#T9" target="_self" title="LATTICE4:th.9">Th9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">b'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">DL</font><b> such that </b><br/><a NAME="E4:20_1"/><i><font color="Green" title="E19">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="c8">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="txt" href="lattice4.html#E1:20"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="lattice4.html#T9" target="_self" title="LATTICE4:th.9">Th9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">c'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">DL</font><b> such that </b><br/><a NAME="E6:20_1"/><i><font color="Green" title="E20">A4</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="c9">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c</font>
 <b>by </b><i><a class="txt" href="lattice4.html#E1:20"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="lattice4.html#T9" target="_self" title="LATTICE4:th.9">Th9</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c4">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">c</font></span>)</span> = 
<font color="Maroon" title="c4">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</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"><font color="Maroon" title="c8">b'</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c9">c'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lattice4.html#E4:20_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="lattice4.html#E6:20_1"><i><font color="Green" title="E20">A4</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"><font color="Maroon" title="c7">a'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">b'</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c9">c'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lattice4.html#E2:20_1"><i><font color="Green" title="E18">A2</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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">b'</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">c'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D11" title="LATTICES:def.11">LATTICES:def 11</a></i>
<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"><font color="Maroon" title="c7">a'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">b'</font></span>)</span></span>)</span> <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"><font color="Maroon" title="c7">a'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">c'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice4.html#D1" target="_self" title="LATTICE4:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">b</font></span>)</span> <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"><font color="Maroon" title="c7">a'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">c'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lattice4.html#E2:20_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="lattice4.html#E4:20_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="lattice4.html#D1" target="_self" title="LATTICE4:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">c</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice4.html#E2:20_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="lattice4.html#E6:20_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="lattice4.html#D1" target="_self" title="LATTICE4:def.1">Def1</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


</div>
