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<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">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="E1:37"/><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>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="funct_6.html#K10">commute</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">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">H</font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:37"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">s'</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> holds  <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">s'</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S5">PBOOLE:sch 5</a>();<br/></i>
<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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s'</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">s'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/><a NAME="E2:37_1"/><b>then </b>
<font color="Maroon">s'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:37_1"/><b>then </b><i><font color="Green">E44</font></i>: 
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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#D15">FUNCT_1:def 15</a></i>;<br/><b>reconsider </b><font color="Maroon">s''</font> = <font color="Maroon">s'</font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/><a NAME="E5:37_1"/>
<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>, , </i>;<br/><a NAME="E6:37_1"/><b>then </b>
 <a href="funct_6.html#K10">commute</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="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="funct_2.html#K1">Funcs</a> <font color="Maroon">I</font>,<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_6.html#T85">FUNCT_6:85</a></i>;<br/><a NAME="E7:37_1"/><b>then </b>
<font color="Maroon">H</font> <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> <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="funct_2.html#K1">Funcs</a> <font color="Maroon">I</font>,<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>, </i>;<br/><b>then consider </b><font color="Maroon">Hs</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:37_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">Hs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s'</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Hs</font> <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> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Hs</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"><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/><a NAME="E10:37_1"/>
<font color="Maroon">s'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E11:37_1"/><b>then </b>
not <span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s'</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D15">FUNCT_1:def 15</a></i>;<br/><a NAME="E12:37_1"/><b>then </b><i><font color="Green">E68</font></i>: 
not the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s'</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><a NAME="E13:37_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">Hs</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_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="E1:37_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Hs</font>
 ;<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="E3:37_1_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</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>  )
 <b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E4:37_1_1"/><i><font color="Green">E74</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s''</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:37_1_1"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Hs</font> &amp; <font color="Maroon">Hs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i'</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_1_1_1"/>
<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"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s''</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">i''</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><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:37_1_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s''</font> <a href="hidden.html#R1">=</a>  <a href="funct_6.html#K10">commute</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>
 <b>by </b><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">i''</font></span>)</span><b> such that </b><br/><a NAME="E5:37_1_1_1"/><i><font color="Green">E77</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">i''</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">i''</font> )
 <b>by </b><i/>;<br/><a NAME="E6:37_1_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i''</font> <a href="hidden.html#R1">=</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="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
 <b>by </b><i>, , , <a class="txt" href="pralg_3.html#E1:37"><i><font color="Green">E38</font></i></a>, <a class="ref" href="pralg_3.html#T1" target="_self">Th1</a>, </i>;<br/><b>consider </b><font color="Maroon">U0</font> being    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E8:37_1_1_1"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">U0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i''</font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s''</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i''</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s''</font> )
 <b>by </b><i><a class="ref" href="pralg_2.html#D16">PRALG_2:def 16</a></i>;<br/><a NAME="E9:37_1_1_1"/><i><font color="Green">E80</font></i>: 
 <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> 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">i''</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s''</font>
 ;<br/><a NAME="E10:37_1_1_1"/>
 <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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Hs</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E11:37_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s'</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E12:37_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<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="txt" href="pralg_3.html#E3:37"><i><font color="Green">E41</font></i></a>, </i>;<br/></div><b>end;</b></div>
<a NAME="E8:37_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s''</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E9:37_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s''</font>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>;<br/>
<b>hence </b><a NAME="E10:37_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s'</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E14:37_1"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s'</font> is   <a href="funct_2.html#NM1">Function</a> of 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>,the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <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>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">H</font> = <font color="Maroon">H</font> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U1</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<b>take </b>
<font color="Maroon">H</font>
;<br/>

<a NAME="E6:37"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">H</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">U1</font>, <a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o</font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D9">MSUALG_3:def 9</a><br/>

<b>assume </b><a NAME="E1:37_2"/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font>;<br/>

<b>set </b><font color="Maroon">s'</font> =  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font>;<br/>
<a NAME="E2:37_2"/><i><font color="Green">E82</font></i>: 
 <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font> <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> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#T10">PRALG_2:10</a></i>;<br/>
<a NAME="E3:37_2"/><b>then </b><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>
 
;<br/>
<b>thus </b><a NAME="E4:37_2"/>
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_2_1_1"/>
(  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_1"><b>suppose </b></a><a NAME="E1:37_2_1_1_1"/><i><font color="Green">E84</font></i>: 
 <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<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> <span class="p3">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span>;<br/><a NAME="E2:37_2_1_1_1"/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="pralg_2.html#T11">PRALG_2:11</a></i>;<br/><a NAME="E3:37_2_1_1_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:37_2_1_1_1"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">H</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E5:37_2_1_1_1"/><i><font color="Green">E87</font></i>: 
 <a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E6:37_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<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> <span class="p3">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><b>then </b><i><font color="Green">E145</font></i>:  <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> <span class="p3">(<span class="default"><span class="p4">(<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> <span class="p4">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#T18">CARD_3:18</a></i>
<br/>.= 
<font color="Maroon">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/><a NAME="E8:37_2_1_1_1"/>
 <a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> <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="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><a href="msualg_1.html#K4">Result</a> <font color="Maroon">o</font>,<span class="p4">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i1</font></span>)</span></span>)</span> where <font color="Olive">i1</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/><b>then consider </b><font color="Maroon">Co</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E10:37_2_1_1_1"/><i><font color="Green">E147</font></i>: 
( <font color="Maroon">Co</font> <a href="hidden.html#R1">=</a>  <a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Co</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Co</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"><a href="msualg_1.html#K4">Result</a> <font color="Maroon">o</font>,<span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i1</font></span>)</span></span>)</span> where <font color="Olive">i1</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i'</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_2_1_1_1_2"/>
<font color="Maroon">i'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><b>then reconsider </b><font color="Maroon">ii</font> = <font color="Maroon">i'</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/><b>consider </b><font color="Maroon">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="Maroon">ii</font></span>)</span><b> such that </b><br/><a NAME="E4:37_2_1_1_1_2"/><i><font color="Green">E148</font></i>: 
( <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ii</font> &amp; <font color="Maroon">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="Maroon">ii</font> )
 <b>by </b><i/>;<br/><a NAME="E5:37_2_1_1_1_2"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">F1</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , </i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p3">(<span class="default"><span class="p4">(<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> <span class="p4">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="pralg_3.html#T1" target="_self">Th1</a>, , </i>
<br/>.= 
 <a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">ii</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_3.html#T1" target="_self">Th1</a>, , <a class="ref" href="msualg_3.html#D9">MSUALG_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div><a NAME="E12:37_2_1_1_1"/><b>then </b><i><font color="Green">E150</font></i>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<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> <span class="p3">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pralg_3.html#K2" target="_self">const</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="pralg_3.html#E1:37"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><a NAME="E13:37_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="pralg_3.html#T1" target="_self">Th1</a>, </i>;<br/><b>hence </b><a NAME="E14:37_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_2"><b>suppose </b></a><a NAME="E1:37_2_1_1_2"/><i><font color="Green">E151</font></i>: 
 <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<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> <span class="p3">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>;<br/><i><font color="Green">E152</font></i>:  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="pralg_2.html#K14">OPS</a> <font color="Maroon">A</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
<br/>.= 
 <a href="cqc_lang.html#K1">IFEQ</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>,<a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_2.html#K2">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3">Frege</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#D20">PRALG_2:def 20</a></i>
<br/>.= 
 <a href="pralg_2.html#K2">Commute</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3">Frege</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/><div><i><font color="Green">E153</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span>;<br/><a NAME="E1:37_2_1_1_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K2">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3">Frege</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E2:37_2_1_1_2_2"/><b>then </b><i><font color="Green">E154</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3">Frege</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="pralg_2.html#D6">PRALG_2:def 6</a></i>;<br/><a NAME="E3:37_2_1_1_2_2"/>
 <a href="funct_6.html#K10">commute</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:37_2_1_1_2_2"/><b>then </b>
 <a href="funct_6.html#K10">commute</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3">Frege</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>hence </b><a NAME="E5:37_2_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3">Frege</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><a NAME="E4:37_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3">Frege</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">f1</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="pralg_2.html#T15">PRALG_2:15</a></i>;<br/><b>set </b><font color="Maroon">D</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">ss</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">ss</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> : verum </span>}</span> ;<br/><a NAME="E6:37_2_1_1_2"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">H</font> <a href="msualg_3.html#K6">#</a> <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"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>,<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"><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">ss</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">ss</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> : verum </span>}</span> </span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E7:37_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<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> <span class="p3">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/><b>then </b><i><font color="Green">E156</font></i>:  <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> <span class="p3">(<span class="default"><span class="p4">(<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> <span class="p4">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#T18">CARD_3:18</a></i>
<br/>.= 
<font color="Maroon">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/><a NAME="E9:37_2_1_1_2"/>
<font color="Maroon">f1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3">Frege</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E10:37_2_1_1_2"/><b>then </b><i><font color="Green">E157</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="pralg_2.html#T16">PRALG_2:16</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i'</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_2_1_1_2_4"/>
<font color="Maroon">i'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><b>then reconsider </b><font color="Maroon">ii</font> = <font color="Maroon">i'</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/><b>consider </b><font color="Maroon">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="Maroon">ii</font></span>)</span><b> such that </b><br/><a NAME="E4:37_2_1_1_2_4"/><i><font color="Green">E158</font></i>: 
( <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ii</font> &amp; <font color="Maroon">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="Maroon">ii</font> )
 <b>by </b><i/>;<br/><a NAME="E5:37_2_1_1_2_4"/><i><font color="Green">E159</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">ii</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="pralg_3.html#T1" target="_self">Th1</a>, <a class="ref" href="msualg_3.html#D9">MSUALG_3:def 9</a></i>;<br/><a NAME="E6:37_2_1_1_2_4"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E7:37_2_1_1_2_4"/><b>then </b><i><font color="Green">E160</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F1</font>
 ;<br/><a NAME="E8:37_2_1_1_2_4"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><i><font color="Green">E162</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K3">Frege</a> <span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p4">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_3.html#D7">MSUALG_3:def 7</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="pralg_1.html#K15">..</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="pralg_2.html#D8">PRALG_2:def 8</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pralg_1.html#D17">PRALG_1:def 17</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_3.html#E3:37"><i><font color="Green">E41</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>
;<br/><a NAME="E10:37_2_1_1_2_4"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/><a NAME="E11:37_2_1_1_2_4"/><b>then </b>
 <a href="funct_6.html#K10">commute</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span> <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"><a href="finseq_1.html#K4">dom</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></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">ss</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">ss</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> : verum </span>}</span> </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">ff</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E13:37_2_1_1_2_4"/><i><font color="Green">E163</font></i>: 
( <font color="Maroon">ff</font> <a href="hidden.html#R1">=</a>  <a href="funct_6.html#K10">commute</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ff</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ff</font> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></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">ss</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">ss</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</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/><a NAME="E14:37_2_1_1_2_4"/>
<font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ii</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ff</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">gg</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E16:37_2_1_1_2_4"/><i><font color="Green">E165</font></i>: 
( <font color="Maroon">gg</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ii</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">gg</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">gg</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">ss</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">ss</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> : verum </span>}</span>  )
 <b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_2_4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_2_1_1_2_4_2"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>
 ;<br/><a NAME="E2:37_2_1_1_2_4_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">s1</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> ;<br/><a NAME="E4:37_2_1_1_2_4_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font>
 ;<br/><a NAME="E5:37_2_1_1_2_4_2"/><b>then </b><i><font color="Green">E167</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</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#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pralg_2.html#T10">PRALG_2:10</a></i>;<br/><b>then </b><i><font color="Green">E168</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">x</font> = 
 <a href="relat_1.html#K1">dom</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#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#T18">CARD_3:18</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#T10">PRALG_2:10</a></i>
;<br/><a NAME="E7:37_2_1_1_2_4_2"/><i><font color="Green">E169</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</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#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E8:37_2_1_1_2_4_2"/><b>then </b>
<font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</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#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="pralg_3.html#T2" target="_self">Th2</a>, , , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E9:37_2_1_1_2_4_2"/><b>then </b><i><font color="Green">E170</font></i>: 
<font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</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">s1</font>
 <b>by </b><i><a class="ref" href="pralg_3.html#T2" target="_self">Th2</a>, , , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><i><font color="Green">E171</font></i>: <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_3.html#T2" target="_self">Th2</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_3.html#T2" target="_self">Th2</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><i><font color="Green">E172</font></i>: <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_3.html#T2" target="_self">Th2</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_3.html#T2" target="_self">Th2</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<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">s1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
;<br/><b>then reconsider </b><font color="Maroon">g</font> = <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ii</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ii</font>
<b>by </b><i>, <a class="ref" href="pralg_3.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_6.html#T86">FUNCT_6:86</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i>, <a class="ref" href="pralg_3.html#T1" target="_self">Th1</a>, , , <a class="ref" href="pralg_3.html#T3" target="_self">Th3</a>, </i>
;<br/></div><b>end;</b></div><a NAME="E18:37_2_1_1_2_4"/><b>then </b>
<font color="Maroon">F1</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ii</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p3">(<span class="default"><span class="p4">(<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> <span class="p4">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font> = 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">ii</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ii</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="pralg_3.html#T1" target="_self">Th1</a>, , </i>
<br/>.= 
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E20:37_2_1_1_2_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p3">(<span class="default"><span class="p4">(<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> <span class="p4">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font>
 ;<br/></div><b>end;</b></div><a NAME="E12:37_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<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> <span class="p3">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font>
 <b>by </b><i>, <a class="txt" href="pralg_3.html#E1:37"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><b>hence </b><a NAME="E13:37_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">U1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E7:37"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> holds  <span class="p1">(<span class="default"><a href="pralg_3.html#K4" target="_self">proj</a> <font color="Maroon">A</font>,<font color="Olive">i</font></span>)</span> <a href="msualg_3.html#K3">**</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/>

<b>consider </b><font color="Maroon">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="Maroon">i</font></span>)</span><b> such that </b><br/><a NAME="E2:37_3"/><i><font color="Green">E173</font></i>: 
( <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">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="Maroon">i</font> )
 <b>by </b><i/>;<br/>
<i><font color="Green">E174</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K4" target="_self">proj</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="msualg_3.html#K3">**</a> <font color="Maroon">H</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K4" target="_self">proj</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">H</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D24">PBOOLE:def 24</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">H</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>

;<br/>
<a NAME="E4:37_3"/><i><font color="Green">E175</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s'</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_3_2"/>
<font color="Maroon">s'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/><b>then reconsider </b><font color="Maroon">s1</font> = <font color="Maroon">s'</font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> ;<br/><a NAME="E3:37_3_2"/>
<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">s1</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">s1</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>, </i>;<br/><a NAME="E4:37_3_2"/><b>then </b>
 <a href="funct_6.html#K10">commute</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">s1</font></span>)</span> <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">s1</font></span>)</span>,<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"><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_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="E6:37_3_2"/><i><font color="Green">E176</font></i>: 
( <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a>  <a href="funct_6.html#K10">commute</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">s1</font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f'</font> <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">s1</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"><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/><a NAME="E7:37_3_2"/><i><font color="Green">E177</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f'</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" target="_self">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s1</font></span>)</span>,<font color="Maroon">i</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_2_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="E1:37_3_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f'</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_3_2_1"/><i><font color="Green">E178</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f'</font> &amp; <font color="Maroon">f'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:37_3_2_1"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<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">s1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s1</font></span>)</span>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E5:37_3_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" target="_self">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s1</font></span>)</span>,<font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><i><font color="Green">E179</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K4" target="_self">proj</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="msualg_3.html#K3">**</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s'</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K4" target="_self">proj</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s1</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="pboole.html#D24">PBOOLE:def 24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_3.html#K3" target="_self">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s1</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_3.html#K3" target="_self">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s1</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<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">s1</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/><a NAME="E9:37_3_2"/><b>then </b><i><font color="Green">E180</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K4" target="_self">proj</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="msualg_3.html#K3">**</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.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="relat_1.html#T46">RELAT_1:46</a></i>;<br/><i><font color="Green">E181</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s'</font></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s1</font></span>)</span>

<br/>.= 
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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_3_2_4"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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>
 ;<br/><a NAME="E2:37_3_2_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<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">s1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s1</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:37_3_2_4"/><b>then </b><i><font color="Green">E183</font></i>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<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">s1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" target="_self">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s1</font></span>)</span>,<font color="Maroon">i</font></span>)</span>
 <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K4" target="_self">proj</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="msualg_3.html#K3">**</a> <font color="Maroon">H</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">x</font> = 
<span class="p1">(<span class="default"><a href="pralg_3.html#K3" target="_self">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">A</font>,<font color="Maroon">s1</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p3">(<span class="default"><span class="p4">(<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">s1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="pralg_3.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p3">(<span class="default"><span class="p4">(<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">s1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="pralg_3.html#T1" target="_self">Th1</a>, </i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:37_3_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K4" target="_self">proj</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="msualg_3.html#K3">**</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s'</font>
 <b>by </b><i>, <a class="txt" href="pralg_3.html#E1:37"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:37_3"/>
<span class="p1">(<span class="default"><a href="pralg_3.html#K4" target="_self">proj</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="msualg_3.html#K3">**</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:37"/>
( <font color="Maroon">H</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">U1</font>, <a href="pralg_2.html#K15">product</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> holds  <span class="p1">(<span class="default"><a href="pralg_3.html#K4" target="_self">proj</a> <font color="Maroon">A</font>,<font color="Olive">i</font></span>)</span> <a href="msualg_3.html#K3">**</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) )
 <b>by </b><i/>;<br/>


</div>
