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

<b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><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="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">I</font>,<font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">U1</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">F</font> be   <a href="pboole.html#NM2">ManySortedFunction</a> of <font color="Maroon">I</font>;<br/>











<b>assume </b><a NAME="E1:34"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>  ex <font color="Olive">F1</font> being  <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U1</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span> st <br/>( <font color="Olive">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">F1</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font> )
 ;<br/>

<b>set </b><font color="Maroon">SU</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>;<br/>
<b>set </b><font color="Maroon">CA</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/>
<b>set </b><font color="Maroon">SA</font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font></span>)</span> where <font color="Olive">i'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">s1</font> is   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon">SA'</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font></span>)</span> where <font color="Olive">i'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">s1</font> is   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon">FS</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font></span>)</span> where <font color="Olive">s1</font> is   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>, <font color="Olive">i'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : verum </span>}</span> ;<br/>
<a NAME="E2:34"/>
<font color="Maroon">F</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">I</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,<span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font></span>)</span> where <font color="Olive">s1</font> is   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>, <font color="Olive">i'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : verum </span>}</span> </span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:34"/><b>then </b>
 <a href="funct_6.html#K10">commute</a> <font color="Maroon">F</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">I</font>,<span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font></span>)</span> where <font color="Olive">s1</font> is   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>, <font color="Olive">i'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : verum </span>}</span> </span>)</span>
 
<b>by </b><i><a class="ref" href="funct_6.html#T85">FUNCT_6:85</a></i>;<br/>
<b>then consider </b><font color="Maroon">F'</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:34"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">F'</font> <a href="hidden.html#R1">=</a>  <a href="funct_6.html#K10">commute</a> <font color="Maroon">F</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F'</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F'</font> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">I</font>,<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font></span>)</span> where <font color="Olive">s1</font> is   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>, <font color="Olive">i'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E6:34"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F'</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">F2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:34"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">F2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F2</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font></span>)</span> where <font color="Olive">s1</font> is   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>, <font color="Olive">i'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : verum </span>}</span>  )
 <b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E9:34"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span>,<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p4">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font></span>)</span> where <font color="Olive">i'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">s1</font> is   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> : verum </span>}</span> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x'</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:34_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">i'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:34_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">i'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i1</font> = <font color="Maroon">i'</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">F'</font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U1</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i1</font></span>)</span><b> such that </b><br/><a NAME="E6:34_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">F'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">F'</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i1</font> )
 <b>by </b><i/>;<br/>
<a NAME="E7:34_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F'</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E8:34_1"/><i><font color="Green">E69</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:34_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font></span>)</span> where <font color="Olive">i'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">s1</font> is   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:34_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E2:34_1_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font></span>)</span> where <font color="Olive">i'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">s1</font> is   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> : verum </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E3:34_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font></span>)</span> where <font color="Olive">i'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">s1</font> is   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> : verum </span>}</span> 
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:34_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font></span>)</span> where <font color="Olive">i'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">s1</font> is   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E11:34_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span>,<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p4">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font></span>)</span> where <font color="Olive">i'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">s1</font> is   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> : verum </span>}</span> </span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:34"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">I</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p5">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s1</font></span>)</span> where <font color="Olive">i'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">s1</font> is   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> : verum </span>}</span> </span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>


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