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<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</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">A</font> being <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b3">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Olive" title="b1">A</font></span>)</span><br/> for <font color="Olive" title="b4">B</font> being  <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Olive" title="b1">A</font>  st <font color="Olive" title="b4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Olive" title="b3">X</font> holds <br/><font color="Olive" title="b4">B</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">(<span class="default"><a href="msualg_8.html#K3" title="MSUALG_8:func.3">EqRelSet</a> <font color="Olive" title="b3">X</font>,<font color="Olive" title="b2">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_5.html#K2" title="MSUALG_5:func.2">EqRelLatt</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Olive" title="b1">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">A</font> be  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>; <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">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span><br/> for <font color="Olive" title="b3">B</font> being  <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Olive" title="b2">X</font> holds <br/><font color="Olive" title="b3">B</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">(<span class="default"><a href="msualg_8.html#K3" title="MSUALG_8:func.3">EqRelSet</a> <font color="Olive" title="b2">X</font>,<font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_5.html#K2" title="MSUALG_5:func.2">EqRelLatt</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>; <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">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span><br/> for <font color="Olive" title="b2">B</font> being  <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Olive" title="b1">X</font> holds <br/><font color="Olive" title="b2">B</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">(<span class="default"><a href="msualg_8.html#K3" title="MSUALG_8:func.3">EqRelSet</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c3">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_5.html#K2" title="MSUALG_5:func.2">EqRelLatt</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span><br/>





<b>set </b><font color="Maroon" title="c4">M</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font>;<br/>
<b>set </b><font color="Maroon" title="c5">E</font> =  <a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font>;<br/>
<b>set </b><font color="Maroon" title="c6">Ei</font> =  <a href="msualg_5.html#K2" title="MSUALG_5:func.2">EqRelLatt</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span>;<br/>
<b>let </b><font color="Maroon" title="c7">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span>; <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">B</font> being  <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Maroon" title="c7">X</font> holds <br/><font color="Olive" title="b1">B</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">(<span class="default"><a href="msualg_8.html#K3" title="MSUALG_8:func.3">EqRelSet</a> <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c3">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_5.html#K2" title="MSUALG_5:func.2">EqRelLatt</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c8">B</font> be   <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Maroon" title="c7">X</font> implies <font color="Maroon" title="c8">B</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">(<span class="default"><a href="msualg_8.html#K3" title="MSUALG_8:func.3">EqRelSet</a> <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c3">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_5.html#K2" title="MSUALG_5:func.2">EqRelLatt</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> )</span><br/>



<b>reconsider </b><font color="Maroon" title="c9">B'</font> = <font color="Maroon" title="c8">B</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span> <b>by </b><i><a class="ref" href="msualg_5.html#D5" title="MSUALG_5:def.5">MSUALG_5:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">Bi</font> = <font color="Maroon" title="c8">B</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c3">i</font> as   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <b>by </b><i><a class="ref" href="msualg_4.html#D3" title="MSUALG_4:def.3">MSUALG_4:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">Bi'</font> = <font color="Maroon" title="c10">Bi</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K2" title="MSUALG_5:func.2">EqRelLatt</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="msualg_7.html#T1" title="MSUALG_7:th.1">MSUALG_7:1</a></i>;<br/>
<b>assume </b><a NAME="E4:24"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Maroon" title="c7">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">B</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">(<span class="default"><a href="msualg_8.html#K3" title="MSUALG_8:func.3">EqRelSet</a> <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c3">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_5.html#K2" title="MSUALG_5:func.2">EqRelLatt</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span></span><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" title="c12">q'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K2" title="MSUALG_5:func.2">EqRelLatt</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">q'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_8.html#K3" title="MSUALG_8:func.3">EqRelSet</a> <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c3">i</font> implies <font color="Maroon" title="c12">q'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">Bi'</font> )</span><br/><b>reconsider </b><font color="Maroon" title="c13">q</font> = <font color="Maroon" title="c12">q'</font> as   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <b>by </b><i><a class="ref" href="msualg_7.html#T1" title="MSUALG_7:th.1">MSUALG_7:1</a></i>;<br/><b>assume </b><a NAME="E2:24_1"/>
<font color="Maroon" title="c12">q'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_8.html#K3" title="MSUALG_8:func.3">EqRelSet</a> <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c3">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">q'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">Bi'</font></span><br/><b>then consider </b><font color="Maroon" title="c14">Eq</font> being   <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E4:24_1"/><i><font color="Green" title="E14">A2</font></i>: 
( <font color="Maroon" title="c12">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Eq</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c14">Eq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> )
 <b>by </b><i><a class="ref" href="msualg_8.html#D3" target="_self" title="MSUALG_8:def.3">Def3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">Eq'</font> = <font color="Maroon" title="c14">Eq</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span> <b>by </b><i><a class="ref" href="msualg_5.html#D5" title="MSUALG_5:def.5">MSUALG_5:def 5</a></i>;<br/><a NAME="E6:24_1"/>
<font color="Maroon" title="c15">Eq'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">B'</font>
 <b>by </b><i><a class="txt" href="#E4:24"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="#E4:24_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="lattice3.html#T38" title="LATTICE3:th.38">LATTICE3:38</a></i>;<br/><a NAME="E7:24_1"/><b>then </b>
<font color="Maroon" title="c14">Eq</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c8">B</font>
 <b>by </b><i><a class="ref" href="msualg_7.html#T7" title="MSUALG_7:th.7">MSUALG_7:7</a></i>;<br/><a NAME="E8:24_1"/><b>then </b>
<font color="Maroon" title="c13">q</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c10">Bi</font>
 <b>by </b><i><a class="txt" href="#E4:24_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/><b>hence </b><a NAME="E9:24_1"/>
<font color="Maroon" title="c12">q'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">Bi'</font>
 <b>by </b><i><a class="ref" href="msualg_8.html#T2" target="_self" title="MSUALG_8:th.2">Th2</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>
<a NAME="E6:24"/><b>then </b><i><font color="Green" title="E14">A3</font></i>: 
 <a href="msualg_8.html#K3" title="MSUALG_8:func.3">EqRelSet</a> <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c3">i</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c11">Bi'</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">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" title="c12">ri</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K2" title="MSUALG_5:func.2">EqRelLatt</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="msualg_8.html#K3" title="MSUALG_8:func.3">EqRelSet</a> <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c3">i</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c12">ri</font> implies <font color="Maroon" title="c11">Bi'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">ri</font> )</span><br/><b>reconsider </b><font color="Maroon" title="c13">ri'</font> = <font color="Maroon" title="c12">ri</font> as   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <b>by </b><i><a class="ref" href="msualg_7.html#T1" title="MSUALG_7:th.1">MSUALG_7:1</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">r'</font> being   <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E3:24_2"/><i><font color="Green" title="E15">A4</font></i>: 
( <font color="Maroon" title="c14">r'</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c13">ri'</font> &amp; ( for <font color="Olive" title="b1">j</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">i</font> holds <br/><font color="Maroon" title="c14">r'</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="msualg_8.html#T16" target="_self" title="MSUALG_8:th.16">Th16</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">r</font> = <font color="Maroon" title="c14">r'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span> <b>by </b><i><a class="ref" href="msualg_5.html#D5" title="MSUALG_5:def.5">MSUALG_5:def 5</a></i>;<br/><b>assume </b><a NAME="E5:24_2"/><i><font color="Green" title="E16">A5</font></i>: 
 <a href="msualg_8.html#K3" title="MSUALG_8:func.3">EqRelSet</a> <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c3">i</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c12">ri</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">Bi'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">ri</font></span><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" title="c16">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c16">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> implies <font color="Maroon" title="c16">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c15">r</font> )</span><br/><b>reconsider </b><font color="Maroon" title="c17">c'</font> = <font color="Maroon" title="c16">c</font> as   <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="msualg_5.html#D5" title="MSUALG_5:def.5">MSUALG_5:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">ci'</font> = <font color="Maroon" title="c17">c'</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c3">i</font> as   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <b>by </b><i><a class="ref" href="msualg_4.html#D3" title="MSUALG_4:def.3">MSUALG_4:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">ci</font> = <font color="Maroon" title="c18">ci'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K2" title="MSUALG_5:func.2">EqRelLatt</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="msualg_7.html#T1" title="MSUALG_7:th.1">MSUALG_7:1</a></i>;<br/><b>assume </b><a NAME="E4:24_2_1"/>
<font color="Maroon" title="c16">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c16">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c15">r</font></span><br/><a NAME="E5:24_2_1"/><b>then </b>
<font color="Maroon" title="c19">ci</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_8.html#K3" title="MSUALG_8:func.3">EqRelSet</a> <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="ref" href="msualg_8.html#D3" target="_self" title="MSUALG_8:def.3">Def3</a></i>;<br/><a NAME="E6:24_2_1"/><b>then </b><i><font color="Green" title="E17">A6</font></i>: 
<font color="Maroon" title="c19">ci</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">ri</font>
 <b>by </b><i><a class="txt" href="#E5:24_2"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">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" title="c20">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c20">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> implies <font color="Maroon" title="c17">c'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">r'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><b>assume </b><a NAME="E1:24_2_1_1"/><i><font color="Green" title="E18">A7</font></i>: 
<font color="Maroon" title="c20">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">c'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">r'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><b>then reconsider </b><font color="Maroon" title="c21">j'</font> = <font color="Maroon" title="c20">j</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ;<br/><b>reconsider </b><font color="Maroon" title="c22">rj'</font> = <font color="Maroon" title="c14">r'</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c21">j'</font>, <font color="Maroon" title="c23">cj'</font> = <font color="Maroon" title="c17">c'</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c21">j'</font> as   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font> <b>by </b><i><a class="ref" href="msualg_4.html#D3" title="MSUALG_4:def.3">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" title="c20">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> or <font color="Maroon" title="c20">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">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" title="c20">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">c'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">r'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><b>hence </b><a NAME="E2:24_2_1_1_1_1"/>
<font color="Maroon" title="c17">c'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">r'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font>
 <b>by </b><i><a class="txt" href="#E3:24_2"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="#E6:24_2_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="msualg_8.html#T2" target="_self" title="MSUALG_8:th.2">Th2</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><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" title="c20">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">c'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">r'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:24_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c14">r'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:24_2"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="#E1:24_2_1_1"><i><font color="Green" title="E18">A7</font></i></a></i>;<br/><a NAME="E3:24_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c23">cj'</font> <a href="eqrel_1.html#K4" title="EQREL_1:func.4">/\</a> <font color="Maroon" title="c22">rj'</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c23">cj'</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/><b>hence </b><a NAME="E4:24_2_1_1_1_2"/>
<font color="Maroon" title="c17">c'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">r'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</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><b>end;</b></div></div><b>end;</b></div><a NAME="E8:24_2_1"/><b>then </b>
<font color="Maroon" title="c17">c'</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c14">r'</font>
 <b>by </b><i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/><b>hence </b><a NAME="E9:24_2_1"/>
<font color="Maroon" title="c16">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c15">r</font>
 <b>by </b><i><a class="ref" href="msualg_7.html#T7" title="MSUALG_7:th.7">MSUALG_7:7</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><a NAME="E7:24_2"/><b>then </b>
<font color="Maroon" title="c7">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c15">r</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a></i>;<br/><a NAME="E8:24_2"/><b>then </b>
<font color="Maroon" title="c9">B'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c15">r</font>
 <b>by </b><i><a class="txt" href="#E4:24"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="lattice3.html#D21" title="LATTICE3:def.21">LATTICE3:def 21</a></i>;<br/><a NAME="E9:24_2"/><b>then </b>
<font color="Maroon" title="c8">B</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c14">r'</font>
 <b>by </b><i><a class="ref" href="msualg_7.html#T7" title="MSUALG_7:th.7">MSUALG_7:7</a></i>;<br/><a NAME="E10:24_2"/><b>then </b>
<font color="Maroon" title="c10">Bi</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c13">ri'</font>
 <b>by </b><i><a class="txt" href="#E3:24_2"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/><b>hence </b><a NAME="E11:24_2"/>
<font color="Maroon" title="c11">Bi'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">ri</font>
 <b>by </b><i><a class="ref" href="msualg_8.html#T2" target="_self" title="MSUALG_8:th.2">Th2</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>
<b>hence </b><a NAME="E8:24"/>
<font color="Maroon" title="c8">B</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">(<span class="default"><a href="msualg_8.html#K3" title="MSUALG_8:func.3">EqRelSet</a> <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c3">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_5.html#K2" title="MSUALG_5:func.2">EqRelLatt</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:24"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="lattice3.html#D21" title="LATTICE3:def.21">LATTICE3:def 21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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