<?xml version="1.0"?>
<div class="add">

<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="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">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/><b>let </b><font color="Maroon" title="c3">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#K6" title="MSUALG_5:func.6">CongrLatt</a> <font color="Maroon" title="c2">A</font></span>)</span>;<br/>





<b>set </b><font color="Maroon" title="c4">M</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font>;<br/>
<a NAME="E1:25"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6" title="MSUALG_5:func.6">CongrLatt</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_lat.html#D16" title="NAT_LAT:def.16">NAT_LAT:def 16</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">X</font> = <font color="Maroon" title="c3">X'</font> as   <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#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">V</font> =  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c6">X</font>,<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#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span> as   <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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="c8">V</font> = <font color="Maroon" title="c7">V</font> as   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">ManySortedRelation</a> of <font color="Maroon" title="c2">A</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c9">V</font> = <font color="Maroon" title="c8">V</font> as  <a href="msualg_4.html#V3" title="MSUALG_4:attr.3">MSEquivalence-like</a> <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">ManySortedRelation</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="msualg_4.html#D5" title="MSUALG_4:def.5">MSUALG_4:def 5</a></i>;<br/>
<a NAME="E6:25"/>
for <font color="Olive" title="b1">s1</font>, <font color="Olive" title="b2">s2</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b3">t</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b3">t</font> <a href="msualg_6.html#R1" title="MSUALG_6:pred.1">is_e.translation_of</a> <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">s1</font>,<font color="Olive" title="b2">s2</font> holds <br/>for <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">a</font>,<font color="Olive" title="b5">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">V</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Olive" title="b1">s1</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">V</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Olive" title="b2">s2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">s1</font>, <font color="Maroon" title="c11">s2</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c12">t</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>



<b>assume </b><a NAME="E1:25_1"/><i><font color="Green" title="E14">A1</font></i>: 
<font color="Maroon" title="c12">t</font> <a href="msualg_6.html#R1" title="MSUALG_6:pred.1">is_e.translation_of</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c10">s1</font>,<font color="Maroon" title="c11">s2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c13">a</font>, <font color="Maroon" title="c14">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E2:25_1"/><i><font color="Green" title="E15">A2</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">V</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c10">s1</font>
 ;<br/>

<a NAME="E3:25_1"/><b>then </b><i><font color="Green" title="E16">A3</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <span class="p1">(<span class="default"><a href="msualg_8.html#K3" title="MSUALG_8:func.3">EqRelSet</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c10">s1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_8.html#T17" target="_self" title="MSUALG_8:th.17">Th17</a></i>;<br/>
<a NAME="E4:25_1"/><i><font color="Green" title="E17">A4</font></i>: 
( <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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="c10">s1</font> &amp; <font color="Maroon" title="c14">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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="c10">s1</font> )
 
<b>by </b><i><a class="txt" href="msualg_8.html#E2:25_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">f</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E6:25_1"/><i><font color="Green" title="E18">A5</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">f</font> &amp; <font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c14">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">f</font></span>)</span> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">f</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="msualg_8.html#K3" title="MSUALG_8:func.3">EqRelSet</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c10">s1</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="msualg_8.html#E3:25_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="msualg_8.html#T10" target="_self" title="MSUALG_8:th.10">Th10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">t'</font> = <font color="Maroon" title="c12">t</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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="c10">s1</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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="c11">s2</font> <b>by </b><i><a class="txt" href="msualg_8.html#E1:25_1"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="msualg_6.html#T11" title="MSUALG_6:th.11">MSUALG_6:11</a></i>;<br/>
<a NAME="E8:25_1"/><i><font color="Green" title="E19">A6</font></i>: 
( <font color="Maroon" title="c16">t'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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="c11">s2</font> &amp; <font color="Maroon" title="c16">t'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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="c11">s2</font> )
 
<b>by </b><i><a class="txt" href="msualg_8.html#E4:25_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<a NAME="E9:25_1"/>
 ex <font color="Olive" title="b1">g</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> &amp; <font color="Maroon" title="c12">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c12">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font></span>)</span> &amp; ( for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="msualg_8.html#K3" title="MSUALG_8:func.3">EqRelSet</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c11">s2</font></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>proof </b></a><div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c12">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c17">g</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E2:25_1_1"/><i><font color="Green" title="E20">A7</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">f</font> &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">f</font></span>)</span> holds <br/><font color="Maroon" title="c17">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></i>
<b>take </b>
<font color="Maroon" title="c17">g</font>
;<br/>

<b>thus </b><a NAME="E3:25_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font>
 <b>by </b><i><a class="txt" href="msualg_8.html#E6:25_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="msualg_8.html#E2:25_1_1"><i><font color="Green" title="E20">A7</font></i></a></i>;<br/>

<a NAME="E4:25_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="msualg_8.html#E6:25_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E5:25_1_1"/>
<font color="Maroon" title="c17">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">a</font>
 <b>by </b><i><a class="txt" href="msualg_8.html#E6:25_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="msualg_8.html#E2:25_1_1"><i><font color="Green" title="E20">A7</font></i></a></i>;<br/>

<a NAME="E6:25_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="msualg_8.html#E6:25_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="msualg_8.html#E2:25_1_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E7:25_1_1"/>
<font color="Maroon" title="c17">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">b</font>
 <b>by </b><i><a class="txt" href="msualg_8.html#E6:25_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="msualg_8.html#E2:25_1_1"><i><font color="Green" title="E20">A7</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c18">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E8:25_1_1"/><i><font color="Green" title="E21">A8</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">j</font> &amp; <font color="Maroon" title="c18">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g</font> )
 ;<br/>

<a NAME="E9:25_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="msualg_8.html#K3" title="MSUALG_8:func.3">EqRelSet</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c10">s1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="msualg_8.html#E6:25_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="msualg_8.html#E2:25_1_1"><i><font color="Green" title="E20">A7</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c19">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:25_1_1"/><i><font color="Green" title="E22">A9</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">Z</font> &amp; <font color="Maroon" title="c19">Z</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="c6">X</font>,<font color="Maroon" title="c10">s1</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">Eq</font> being   <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E13:25_1_1"/><i><font color="Green" title="E23">A10</font></i>: 
( <font color="Maroon" title="c19">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Eq</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c10">s1</font> &amp; <font color="Maroon" title="c20">Eq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> )
 <b>by </b><i><a class="txt" href="msualg_8.html#E11:25_1_1"><i><font color="Green" title="E22">A9</font></i></a>, <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="c21">Eq</font> = <font color="Maroon" title="c20">Eq</font> as   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">ManySortedRelation</a> of <font color="Maroon" title="c2">A</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c22">Eq</font> = <font color="Maroon" title="c21">Eq</font> as  <a href="msualg_4.html#V3" title="MSUALG_4:attr.3">MSEquivalence-like</a> <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">ManySortedRelation</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="msualg_4.html#D5" title="MSUALG_4:def.5">MSUALG_4:def 5</a></i>;<br/>
<a NAME="E16:25_1_1"/>
<font color="Maroon" title="c22">Eq</font> is   <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="msualg_8.html#E13:25_1_1"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="msualg_5.html#D6" title="MSUALG_5:def.6">MSUALG_5:def 6</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c23">Eq</font> = <font color="Maroon" title="c22">Eq</font> as  <a href="msualg_4.html#V3" title="MSUALG_4:attr.3">MSEquivalence-like</a> <a href="msualg_6.html#V2" title="MSUALG_6:attr.2">compatible</a> <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">ManySortedRelation</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="msualg_6.html#T27" title="MSUALG_6:th.27">MSUALG_6:27</a></i>;<br/>
<a NAME="E18:25_1_1"/>
 ex <font color="Olive" title="b1">W</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</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="c6">X</font>,<font color="Maroon" title="c11">s2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon" title="c24">W</font> = <font color="Maroon" title="c23">Eq</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c11">s2</font>;<br/>

<b>thus </b><a NAME="E1:25_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">W</font>
 <b>by </b><i><a class="txt" href="msualg_8.html#E1:25_1"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="msualg_8.html#E11:25_1_1"><i><font color="Green" title="E22">A9</font></i></a>, <a class="txt" href="msualg_8.html#E13:25_1_1"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="msualg_6.html#D8" title="MSUALG_6:def.8">MSUALG_6:def 8</a></i>;<br/>

<b>thus </b><a NAME="E2:25_1_1_1"/>
<font color="Maroon" title="c24">W</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="c6">X</font>,<font color="Maroon" title="c11">s2</font>
 <b>by </b><i><a class="txt" href="msualg_8.html#E13:25_1_1"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="msualg_8.html#D3" target="_self" title="MSUALG_8:def.3">Def3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:25_1_1"/><b>then </b><i><font color="Green" title="E24">A11</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="msualg_8.html#K3" title="MSUALG_8:func.3">EqRelSet</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c11">s2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E20:25_1_1"/>
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="msualg_8.html#E2:25_1_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="msualg_8.html#E8:25_1_1"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E21:25_1_1"/><b>then </b><i><font color="Green" title="E25">A12</font></i>: 
<font color="Maroon" title="c17">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="msualg_8.html#E2:25_1_1"><i><font color="Green" title="E20">A7</font></i></a></i>;<br/>
<a NAME="E22:25_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">f</font> )
 
<b>by </b><i><a class="txt" href="msualg_8.html#E2:25_1_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="msualg_8.html#E8:25_1_1"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E23:25_1_1"/><b>then </b>
<font color="Maroon" title="c18">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E24:25_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="msualg_8.html#K3" title="MSUALG_8:func.3">EqRelSet</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c11">s2</font></span>)</span>
 <b>by </b><i><a class="txt" href="msualg_8.html#E2:25_1_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="msualg_8.html#E19:25_1_1"><i><font color="Green" title="E24">A11</font></i></a>, <a class="txt" href="msualg_8.html#E21:25_1_1"><i><font color="Green" title="E25">A12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:25_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <span class="p1">(<span class="default"><a href="msualg_8.html#K3" title="MSUALG_8:func.3">EqRelSet</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c11">s2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="msualg_8.html#E8:25_1"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="msualg_8.html#T10" target="_self" title="MSUALG_8:th.10">Th10</a></i>;<br/>
<b>hence </b><a NAME="E11:25_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">V</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c11">s2</font>
 <b>by </b><i><a class="ref" href="msualg_8.html#T17" target="_self" title="MSUALG_8:th.17">Th17</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c10">V</font> = <font color="Maroon" title="c9">V</font> as  <a href="msualg_4.html#V3" title="MSUALG_4:attr.3">MSEquivalence-like</a> <a href="msualg_6.html#V3" title="MSUALG_6:attr.3">invariant</a> <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">ManySortedRelation</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="msualg_6.html#D8" title="MSUALG_6:def.8">MSUALG_6:def 8</a></i>;<br/>
<a NAME="E8:25"/>
<font color="Maroon" title="c10">V</font> is <a href="msualg_6.html#V2" title="MSUALG_6:attr.2">compatible</a>
 
;<br/>
<b>hence </b><a NAME="E9:25"/>
 <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c3">X'</font>,<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#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span> is   <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="msualg_6.html#T27" title="MSUALG_6:th.27">MSUALG_6:27</a></i>;<br/>


</div>
