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<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">A</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span>)</span>;<br/>





<b>set </b><font color="Maroon">Z</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span>)</span> : ( <font color="Olive">x</font> is   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">A</font> &amp; <font color="Maroon">C</font> <a href="lattices.html#R3">[=</a> <font color="Olive">x</font> ) </span>}</span> ;<br/>
<b>assume </b><a NAME="E1:17"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">C</font> is   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">C'</font> = <font color="Maroon">C</font> as   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">A</font> ;<br/>
<a NAME="E3:17"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span>)</span> : ( <font color="Olive">x</font> is   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">A</font> &amp; <font color="Maroon">C</font> <a href="lattices.html#R3">[=</a> <font color="Olive">x</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span>)</span>;<br/><b>assume </b><a NAME="E1:17_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span>)</span> : ( <font color="Olive">x</font> is   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">A</font> &amp; <font color="Maroon">C</font> <a href="lattices.html#R3">[=</a> <font color="Olive">x</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span>)</span><b> such that </b><br/><a NAME="E3:17_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> is   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">A</font> &amp; <font color="Maroon">C</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> )
 ;<br/><b>thus </b><a NAME="E4:17_1"/>
<font color="Maroon">C</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E5:17"/><b>then </b>
<font color="Maroon">C</font> <a href="lattice3.html#R3">is_less_than</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span>)</span> : ( <font color="Olive">x</font> is   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">A</font> &amp; <font color="Maroon">C</font> <a href="lattices.html#R3">[=</a> <font color="Olive">x</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/>
<a NAME="E6:17"/><b>then </b>
<font color="Maroon">C'</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span>)</span> : ( <font color="Olive">x</font> is   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">A</font> &amp; <font color="Maroon">C</font> <a href="lattices.html#R3">[=</a> <font color="Olive">x</font> ) </span>}</span> ,<span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#T42">LATTICE3:42</a></i>;<br/>
<b>hence </b><a NAME="E7:17"/>
 <a href="msualg_8.html#K1" target="_self">CongrCl</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 ;<br/>


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