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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">IL</font> be  <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</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">IL</font>,<font color="Maroon" title="c1">CL</font>;<br/>





<b>assume </b><a NAME="E1:38"/>
<font color="Maroon" title="c3">f</font> is <a href="lattice4.html#V3" title="LATTICE4:attr.3">isomorphism</a>
 ;<br/>

<a NAME="E2:38"/><b>then </b><i><font color="Green" title="E37">A1</font></i>: 
( <font color="Maroon" title="c3">f</font> is <a href="lattice4.html#V2" title="LATTICE4:attr.2">epimorphism</a> &amp; <font color="Maroon" title="c3">f</font> is <a href="lattice4.html#V1" title="LATTICE4:attr.1">monomorphism</a> )
 
<b>by </b><i><a class="ref" href="lattice4.html#D4" target="_self" title="LATTICE4:def.4">Def4</a></i>;<br/>
<b>thus </b><a NAME="E3:38"/>
<font color="Maroon" title="c1">CL</font> is <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">CL</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D7" title="FILTER_0:def.7">FILTER_0:def 7</a><br/>

<b>consider </b><font color="Maroon" title="c6">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">IL</font><b> such that </b><br/><a NAME="E2:38_1"/><i><font color="Green" title="E38">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="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="txt" href="lattice4.html#E2:38"><i><font color="Green" title="E37">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="c7">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">IL</font><b> such that </b><br/><a NAME="E4:38_1"/><i><font color="Green" title="E39">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="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font>
 <b>by </b><i><a class="txt" href="lattice4.html#E2:38"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="lattice4.html#T9" target="_self" title="LATTICE4:th.9">Th9</a></i>;<br/>
<b>take </b><font color="Maroon" title="c8">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"><font color="Maroon" title="c6">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c7">j</font></span>)</span>;<br/>

<b>thus </b><a NAME="E5:38_1"/>
<font color="Maroon" title="c4">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">q</font>
 <b>by </b><i><a class="txt" href="lattice4.html#E2:38_1"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="lattice4.html#E4:38_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="lattice4.html#T33" target="_self" title="LATTICE4:th.33">Th33</a></i>;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c9">r1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">CL</font>;<br/><b>consider </b><font color="Maroon" title="c10">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">IL</font><b> such that </b><br/><a NAME="E2:38_1_1"/><i><font color="Green" title="E40">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="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r1</font>
 <b>by </b><i><a class="txt" href="lattice4.html#E2:38"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="lattice4.html#T9" target="_self" title="LATTICE4:th.9">Th9</a></i>;<br/><b>assume </b><a NAME="E3:38_1_1"/>
<font color="Maroon" title="c4">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">r1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">q</font>
 ;<br/><b>hence </b><a NAME="E4:38_1_1"/>
<font color="Maroon" title="c9">r1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">r</font>
 <b>by </b><i><a class="txt" href="lattice4.html#E2:38"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="lattice4.html#E2:38_1"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="lattice4.html#E4:38_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="lattice4.html#E2:38_1_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="lattice4.html#T34" target="_self" title="LATTICE4:th.34">Th34</a></i>;<br/></div>
<b>end;</b></div>


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

<b>then reconsider </b><font color="Maroon" title="c4">CL</font> = <font color="Maroon" title="c1">CL</font> as  <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</a> <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a> ;<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">IL</font>,<font color="Maroon" title="c4">CL</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c7">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">IL</font>;<br/><a NAME="E1:38_2"/><i><font color="Green" title="E38">A5</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> <font color="Maroon" title="c6">i</font></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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">j</font>
 <b>by </b><i><a class="ref" href="lattice4.html#T33" target="_self" title="LATTICE4:th.33">Th33</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">r1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">CL</font>;<br/><b>consider </b><font color="Maroon" title="c9">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">IL</font><b> such that </b><br/><a NAME="E2:38_2_1"/><i><font color="Green" title="E39">A6</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">r1</font>
 <b>by </b><i><a class="txt" href="lattice4.html#E2:38"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="lattice4.html#T9" target="_self" title="LATTICE4:th.9">Th9</a></i>;<br/><b>assume </b><a NAME="E3:38_2_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">i</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">r1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">j</font>
 ;<br/><b>hence </b><a NAME="E4:38_2_1"/>
<font color="Maroon" title="c8">r1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</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">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c7">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice4.html#E2:38"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="lattice4.html#E2:38_2_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="lattice4.html#T34" target="_self" title="LATTICE4:th.34">Th34</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:38_2"/>
<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">i</font></span>)</span> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</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="c7">j</font></span>)</span> <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">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c7">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice4.html#E1:38_2"><i><font color="Green" title="E38">A5</font></i></a>, <a class="ref" href="filter_0.html#D8" title="FILTER_0:def.8">FILTER_0:def 8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:38"/>
<font color="Maroon" title="c3">f</font> <a href="lattice4.html#R1" title="LATTICE4:pred.1">preserves_implication</a> 
 <b>by </b><i><a class="ref" href="lattice4.html#D6" target="_self" title="LATTICE4:def.6">Def6</a></i>;<br/>


</div>
