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

<b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">A</font> be    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">I</font>,<font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">o</font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>;<br/>







<b>assume </b><a NAME="E1:12"/><i><font color="Green">E38</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/>

<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"><a href="pralg_2.html#K12">OPER</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>;<br/>
<a NAME="E2:12"/>
 <a href="funct_6.html#K10">commute</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K12">OPER</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">I</font>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <span class="p4">(<span class="default"><a href="pralg_2.html#K12">OPER</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#T13">PRALG_2:13</a></i>;<br/>
<b>then consider </b><font color="Maroon">f1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:12"/><i><font color="Green">E41</font></i>: 
 <a href="funct_6.html#K10">commute</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K12">OPER</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font>
 <b>and </b><br/><a NAME="E5:12"/><i><font color="Green">E43</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E6:12"/><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</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="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <span class="p3">(<span class="default"><a href="pralg_2.html#K12">OPER</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E7:12"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K12">OPER</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K12">OPER</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 
<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">fb</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:12"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">fb</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"><a href="pralg_2.html#K12">OPER</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">fb</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K12">OPER</a> <font color="Maroon">A</font></span>)</span></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">C</font> =  <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">i'</font></span>)</span></span>)</span> where <font color="Olive">i'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : verum </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><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="E1:12_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p3">(<span class="default"><a href="pralg_2.html#K12">OPER</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span>
 ;<br/><a NAME="E2:12_1"/><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K12">OPER</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font>
 ;<br/><b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:12_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p3">(<span class="default"><a href="pralg_2.html#K12">OPER</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span> &amp; <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"><a href="pralg_2.html#K12">OPER</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i>, , <a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a></i>;<br/><i><font color="Green">E74</font></i>: <font color="Maroon">x'</font> = 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i/>
<br/>.= 
 <a href="msualg_1.html#K5">Den</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">a</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#T14">PRALG_2:14</a></i>
;<br/><b>then </b><i><font color="Green">E75</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">x'</font> = 
 <a href="msualg_1.html#K3">Args</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">a</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:12_1_3"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">x'</font>
 ;<br/><b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:12_1_3"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">x'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:12_1_3"/>
<font color="Maroon">x'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</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"><font color="Maroon">A</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:12_1_3"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">c</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K4">Result</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">a</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E6:12_1_3"/>
 <a href="msualg_1.html#K4">Result</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">a</font></span>)</span> <a href="hidden.html#R2">in</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">i'</font></span>)</span></span>)</span> where <font color="Olive">i'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : verum </span>}</span> 
 ;<br/><b>hence </b><a NAME="E7:12_1_3"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><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">i'</font></span>)</span></span>)</span> where <font color="Olive">i'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : verum </span>}</span> 
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><a NAME="E10:12_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">x'</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">i'</font></span>)</span></span>)</span> where <font color="Olive">i'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E11:12_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</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>,<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">i'</font></span>)</span></span>)</span> where <font color="Olive">i'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : verum </span>}</span> </span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:12"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p3">(<span class="default"><a href="pralg_2.html#K12">OPER</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</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>,<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">i'</font></span>)</span></span>)</span> where <font color="Olive">i'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : verum </span>}</span> </span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E12:12"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K12">OPER</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</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"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></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"><a href="msualg_1.html#K4">Result</a> <font color="Maroon">o</font>,<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></span>)</span> where <font color="Olive">i'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : verum </span>}</span> </span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>


</div>
