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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">i</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/>





<b>set </b><font color="Maroon">M</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>;<br/>
<b>set </b><font color="Maroon">E</font> =  <a href="msualg_5.html#K5">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>;<br/>
<b>set </b><font color="Maroon">Ei</font> =  <a href="msualg_5.html#K2">EqRelLatt</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>;<br/>
<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</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>let </b><font color="Maroon">B</font> be   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>;<br/>



<b>reconsider </b><font color="Maroon">B'</font> = <font color="Maroon">B</font> as   <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>by </b><i><a class="ref" href="msualg_5.html#D5">MSUALG_5:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Bi</font> = <font color="Maroon">B</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">i</font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <b>by </b><i><a class="ref" href="msualg_4.html#D3">MSUALG_4:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Bi'</font> = <font color="Maroon">Bi</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K2">EqRelLatt</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="msualg_7.html#T1">MSUALG_7:1</a></i>;<br/>
<b>assume </b><a NAME="E4:24"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Maroon">X</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_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#K2">EqRelLatt</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q'</font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <b>by </b><i><a class="ref" href="msualg_7.html#T1">MSUALG_7:1</a></i>;<br/><b>assume </b><a NAME="E2:24_1"/>
<font color="Maroon">q'</font> <a href="hidden.html#R2">in</a>  <a href="msualg_8.html#K3" target="_self">EqRelSet</a> <font color="Maroon">X</font>,<font color="Maroon">i</font>
 ;<br/><b>then consider </b><font color="Maroon">Eq</font> being   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E4:24_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">q'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Eq</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">Eq</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">Eq'</font> = <font color="Maroon">Eq</font> as   <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>by </b><i><a class="ref" href="msualg_5.html#D5">MSUALG_5:def 5</a></i>;<br/><a NAME="E6:24_1"/>
<font color="Maroon">Eq'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">B'</font>
 <b>by </b><i>, , <a class="ref" href="lattice3.html#T38">LATTICE3:38</a></i>;<br/><a NAME="E7:24_1"/><b>then </b>
<font color="Maroon">Eq</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="msualg_7.html#T7">MSUALG_7:7</a></i>;<br/><a NAME="E8:24_1"/><b>then </b>
<font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Bi</font>
 <b>by </b><i>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/><b>hence </b><a NAME="E9:24_1"/>
<font color="Maroon">q'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">Bi'</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E6:24"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="msualg_8.html#K3" target="_self">EqRelSet</a> <font color="Maroon">X</font>,<font color="Maroon">i</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">Bi'</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">ri</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K2">EqRelLatt</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">ri'</font> = <font color="Maroon">ri</font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <b>by </b><i><a class="ref" href="msualg_7.html#T1">MSUALG_7:1</a></i>;<br/><b>consider </b><font color="Maroon">r'</font> being   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:24_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">r'</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ri'</font> &amp; ( for <font color="Olive">j</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>  st <font color="Olive">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> holds <br/><font color="Maroon">r'</font> <a href="msualg_4.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K1">nabla</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span> ) )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r'</font> as   <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>by </b><i><a class="ref" href="msualg_5.html#D5">MSUALG_5:def 5</a></i>;<br/><b>assume </b><a NAME="E5:24_2"/><i><font color="Green">E63</font></i>: 
 <a href="msualg_8.html#K3" target="_self">EqRelSet</a> <font color="Maroon">X</font>,<font color="Maroon">i</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">ri</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_1"><b>now </b></a><div class="add"><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>reconsider </b><font color="Maroon">c'</font> = <font color="Maroon">c</font> as   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="msualg_5.html#D5">MSUALG_5:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">ci'</font> = <font color="Maroon">c'</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">i</font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <b>by </b><i><a class="ref" href="msualg_4.html#D3">MSUALG_4:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">ci</font> = <font color="Maroon">ci'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K2">EqRelLatt</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="msualg_7.html#T1">MSUALG_7:1</a></i>;<br/><b>assume </b><a NAME="E4:24_2_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E5:24_2_1"/><b>then </b>
<font color="Maroon">ci</font> <a href="hidden.html#R2">in</a>  <a href="msualg_8.html#K3" target="_self">EqRelSet</a> <font color="Maroon">X</font>,<font color="Maroon">i</font>
 <b>by </b><i/>;<br/><a NAME="E6:24_2_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">ci</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">ri</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:24_2_1_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/><b>then reconsider </b><font color="Maroon">j'</font> = <font color="Maroon">j</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ;<br/><b>reconsider </b><font color="Maroon">rj'</font> = <font color="Maroon">r'</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">j'</font>, <font color="Maroon">cj'</font> = <font color="Maroon">c'</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">j'</font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <b>by </b><i><a class="ref" href="msualg_4.html#D3">MSUALG_4:def 3</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_2_1_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> or <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:24_2_1_1_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:24_2_1_1_1_1"/>
<font color="Maroon">c'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">r'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:24_2_1_1_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><a NAME="E2:24_2_1_1_1_2"/><b>then </b>
<font color="Maroon">r'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K1">nabla</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:24_2_1_1_1_2"/><b>then </b>
<font color="Maroon">cj'</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">rj'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cj'</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><b>hence </b><a NAME="E4:24_2_1_1_1_2"/>
<font color="Maroon">c'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">r'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E8:24_2_1"/><b>then </b>
<font color="Maroon">c'</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">r'</font>
 <b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/><b>hence </b><a NAME="E9:24_2_1"/>
<font color="Maroon">c</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="msualg_7.html#T7">MSUALG_7:7</a></i>;<br/></div><b>end;</b></div><a NAME="E7:24_2"/><b>then </b>
<font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a></i>;<br/><a NAME="E8:24_2"/><b>then </b>
<font color="Maroon">B'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/><a NAME="E9:24_2"/><b>then </b>
<font color="Maroon">B</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">r'</font>
 <b>by </b><i><a class="ref" href="msualg_7.html#T7">MSUALG_7:7</a></i>;<br/><a NAME="E10:24_2"/><b>then </b>
<font color="Maroon">Bi</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">ri'</font>
 <b>by </b><i>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/><b>hence </b><a NAME="E11:24_2"/>
<font color="Maroon">Bi'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">ri</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:24"/>
<font color="Maroon">B</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">(<span class="default"><a href="msualg_8.html#K3" target="_self">EqRelSet</a> <font color="Maroon">X</font>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_5.html#K2">EqRelLatt</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/>


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