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

<b>set </b><font color="Maroon" title="c4">X</font> = the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">S</font>;<br/>
<b>set </b><font color="Maroon" title="c5">AS</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c2">S</font>;<br/>
<b>set </b><font color="Maroon" title="c6">RS</font> = <span class="p1">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">S</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">o</font> being  <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c2">S</font>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">o</font> holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">IFEQ</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b1">o</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Olive" title="b1">o</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_2.html#K2" title="PRALG_2:func.2">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" title="PRALG_2:func.3">Frege</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Olive" title="b1">o</font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E1:38_1_1"/><i><font color="Green" title="E18">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">S</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">S</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">o</font> = <font color="Maroon" title="c7">x</font> as   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<b>take </b>
 <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">IFEQ</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c8">o</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c8">o</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_2.html#K2" title="PRALG_2:func.2">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" title="PRALG_2:func.3">Frege</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c8">o</font></span>)</span></span>)</span></span>)</span>
;<br/>

<b>let </b><font color="Maroon" title="c9">o1</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c2">S</font>;<br/>

<b>assume </b><a NAME="E3:38_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">o1</font>
 ;<br/>

<b>hence </b><a NAME="E4:38_1_1_1"/>
 <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">IFEQ</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c8">o</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c8">o</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_2.html#K2" title="PRALG_2:func.2">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" title="PRALG_2:func.3">Frege</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c8">o</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">IFEQ</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c9">o1</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c9">o1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_2.html#K2" title="PRALG_2:func.2">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" title="PRALG_2:func.3">Frege</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c9">o1</font></span>)</span></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c7">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:38_1_1"/><i><font color="Green" title="E19">A2</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">S</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">S</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="classes1.html#S1" title="CLASSES1:sch.1">CLASSES1:sch 1</a>(<a class="txt" href="pralg_2.html#E1:38_1_1"><i><font color="Green" title="E18">A1</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c7">f</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:38_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">f</font> holds <br/><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_1_1_2"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">f</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c10">o</font> = <font color="Maroon" title="c9">x</font> as   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1"><i><font color="Green" title="E19">A2</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1_1_2_1"/>
(  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:38_1_1_2_1_1"/><i><font color="Green" title="E21">A4</font></i>: 
 <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> = 
 <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">IFEQ</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">o</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_2.html#K2" title="PRALG_2:func.2">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" title="PRALG_2:func.3">Frege</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="pralg_2.html#E1:38_1_1_2"><i><font color="Green" title="E20">A3</font></i></a></i>
<br/>.= 
 <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c10">o</font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_2_1_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
;<br/><b>hence </b><a NAME="E3:38_1_1_2_1_1"/>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:38_1_1_2_1_2"/><i><font color="Green" title="E21">A5</font></i>: 
 <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> = 
 <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">IFEQ</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">o</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_2.html#K2" title="PRALG_2:func.2">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" title="PRALG_2:func.3">Frege</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="pralg_2.html#E1:38_1_1_2"><i><font color="Green" title="E20">A3</font></i></a></i>
<br/>.= 
 <a href="pralg_2.html#K2" title="PRALG_2:func.2">Commute</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3" title="PRALG_2:func.3">Frege</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_2_1_2"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
;<br/><b>hence </b><a NAME="E3:38_1_1_2_1_2"/>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c9">f</font> = <font color="Maroon" title="c8">f</font> as   <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="ref" href="funcop_1.html#D6" title="FUNCOP_1:def.6">FUNCOP_1:def 6</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:38_1_1_3"/>
<font color="Maroon" title="c1">I</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c10">I'</font> = <font color="Maroon" title="c1">I</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>reconsider </b><font color="Maroon" title="c11">A'</font> = <font color="Maroon" title="c3">A</font> as    <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c10">I'</font>,<font color="Maroon" title="c2">S</font> ;<br/><a NAME="E4:38_1_1_3"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">S</font> holds <br/><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_1_1_3_1"/><i><font color="Green" title="E20">A6</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">S</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c13">o</font> = <font color="Maroon" title="c12">x</font> as   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<b>set </b><font color="Maroon" title="c14">C</font> =  <a href="pralg_2.html#K2" title="PRALG_2:func.2">Commute</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3" title="PRALG_2:func.3">Frege</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c15">F</font> =  <a href="pralg_2.html#K3" title="PRALG_2:func.3">Frege</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c16">Ar</font> = the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c2">S</font>;<br/>
<b>set </b><font color="Maroon" title="c17">Rs</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">S</font>;<br/>
<b>set </b><font color="Maroon" title="c18">Cr</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>;<br/>
<b>set </b><font color="Maroon" title="c19">co</font> =  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span>;<br/>
<a NAME="E3:38_1_1_3_1"/><i><font color="Green" title="E21">A7</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">S</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E4:38_1_1_3_1"/><b>then </b>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>then </b><i><font color="Green" title="E22">A8</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">o</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_3_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="pralg_2.html#E3:38_1_1_3_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_1.html#D6" title="MSUALG_1:def.6">MSUALG_1:def 6</a></i>
<br/>.= 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D19" title="PBOOLE:def.19">PBOOLE:def 19</a></i>
;<br/>
<b>set </b><font color="Maroon" title="c20">ar</font> =  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font>;<br/>
<a NAME="E6:38_1_1_3_1"/><i><font color="Green" title="E23">A9</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E7:38_1_1_3_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E8:38_1_1_3_1"/><b>then </b><i><font color="Green" title="E24">A10</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E6:38_1_1_3_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E9:38_1_1_3_1"/><i><font color="Green" title="E25">A11</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">S</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E10:38_1_1_3_1"/><b>then </b>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>then </b><i><font color="Green" title="E26">A12</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">o</font> = 
<span class="p1">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_3_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="pralg_2.html#E9:38_1_1_3_1"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c13">o</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_1.html#D7" title="MSUALG_1:def.7">MSUALG_1:def 7</a></i>
<br/>.= 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#D17" target="_self" title="PRALG_2:def.17">Def17</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_3_1_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1_1_3_1_3"/>
(  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_3_1_3_1"><b>suppose </b></a><a NAME="E1:38_1_1_3_1_3_1"/><i><font color="Green" title="E27">A13</font></i>: 
 <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><i><font color="Green" title="E28">A14</font></i>: <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> = 
 <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">IFEQ</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_2.html#K2" title="PRALG_2:func.2">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" title="PRALG_2:func.3">Frege</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="pralg_2.html#E1:38_1_1_3_1"><i><font color="Green" title="E20">A6</font></i></a></i>
<br/>.= 
 <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_1"><i><font color="Green" title="E27">A13</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
;<br/><a NAME="E3:38_1_1_3_1_3_1"/><i><font color="Green" title="E29">A15</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="pralg_2.html#E5:38_1_1_3_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_1"><i><font color="Green" title="E27">A13</font></i></a>, <a class="ref" href="card_3.html#T19" title="CARD_3:th.19">CARD_3:19</a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></i>;<br/><a NAME="E4:38_1_1_3_1_3_1"/><i><font color="Green" title="E30">A16</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><b>set </b><font color="Maroon" title="c21">rs</font> =  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c13">o</font>;<br/><a NAME="E5:38_1_1_3_1_3_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="pralg_2.html#K8" title="PRALG_2:func.8">|.</a><span class="default"><font color="Maroon" title="c11">A'</font></span><a href="pralg_2.html#K8" title="PRALG_2:func.8">.|</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3_1_3_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:38_1_1_3_1_3_1_2"/>
<font color="Maroon" title="c22">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c23">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:38_1_1_3_1_3_1_2"/><i><font color="Green" title="E31">A17</font></i>: 
( <font color="Maroon" title="c23">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">i</font> = <font color="Maroon" title="c23">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">I'</font> <b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_1_2"><i><font color="Green" title="E31">A17</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:38_1_1_3_1_3_1_2"/><i><font color="Green" title="E32">A18</font></i>: 
<font color="Maroon" title="c22">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c13">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_1_2"><i><font color="Green" title="E31">A17</font></i></a>, <a class="ref" href="pralg_2.html#T14" target="_self" title="PRALG_2:th.14">Th14</a></i>;<br/>
<a NAME="E6:38_1_1_3_1_3_1_2"/><i><font color="Green" title="E33">A19</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c13">o</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c13">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">i</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c13">o</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <font color="Maroon" title="c13">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">i</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E7:38_1_1_3_1_3_1_2"/><i><font color="Green" title="E34">A20</font></i>: 
 <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c13">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_1"><i><font color="Green" title="E27">A13</font></i></a>, <a class="ref" href="pralg_2.html#T11" target="_self" title="PRALG_2:th.11">Th11</a></i>;<br/>
<b>set </b><font color="Maroon" title="c25">So</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">i</font></span>)</span>;<br/>
<a NAME="E8:38_1_1_3_1_3_1_2"/><i><font color="Green" title="E35">A21</font></i>: 
 <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <font color="Maroon" title="c13">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c13">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#T10" target="_self" title="PRALG_2:th.10">Th10</a></i>;<br/>
<a NAME="E9:38_1_1_3_1_3_1_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E10:38_1_1_3_1_3_1_2"/><b>then </b>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E11:38_1_1_3_1_3_1_2"/><b>then </b>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E12:38_1_1_3_1_3_1_2"/><b>then </b><i><font color="Green" title="E36">A22</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c13">o</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="pralg_2.html#K7" title="PRALG_2:func.7">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="pralg_2.html#K7" title="PRALG_2:func.7">.|</a></span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E6:38_1_1_3_1_3_1_2"><i><font color="Green" title="E33">A19</font></i></a>, <a class="txt" href="pralg_2.html#E8:38_1_1_3_1_3_1_2"><i><font color="Green" title="E35">A21</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E13:38_1_1_3_1_3_1_2"/>
<span class="p1"><a href="pralg_2.html#K7" title="PRALG_2:func.7">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="pralg_2.html#K7" title="PRALG_2:func.7">.|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="pralg_2.html#K7" title="PRALG_2:func.7">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b1">d</font></span>)</span></span><a href="pralg_2.html#K7" title="PRALG_2:func.7">.|</a></span> where <font color="Olive" title="b1">d</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">I'</font> : verum </span>}</span> 
 
;<br/>
<a NAME="E14:38_1_1_3_1_3_1_2"/><b>then </b>
<span class="p1"><a href="pralg_2.html#K7" title="PRALG_2:func.7">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="pralg_2.html#K7" title="PRALG_2:func.7">.|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2"><a href="pralg_2.html#K7" title="PRALG_2:func.7">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b1">d</font></span>)</span></span><a href="pralg_2.html#K7" title="PRALG_2:func.7">.|</a></span> where <font color="Olive" title="b1">d</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">I'</font> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E15:38_1_1_3_1_3_1_2"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c13">o</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="pralg_2.html#K8" title="PRALG_2:func.8">|.</a><span class="default"><font color="Maroon" title="c11">A'</font></span><a href="pralg_2.html#K8" title="PRALG_2:func.8">.|</a></span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E12:38_1_1_3_1_3_1_2"><i><font color="Green" title="E36">A22</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E16:38_1_1_3_1_3_1_2"/>
<font color="Maroon" title="c22">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="pralg_2.html#K8" title="PRALG_2:func.8">|.</a><span class="default"><font color="Maroon" title="c11">A'</font></span><a href="pralg_2.html#K8" title="PRALG_2:func.8">.|</a></span>
 <b>by </b><i><a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_1_2"><i><font color="Green" title="E32">A18</font></i></a>, <a class="txt" href="pralg_2.html#E6:38_1_1_3_1_3_1_2"><i><font color="Green" title="E33">A19</font></i></a>, <a class="txt" href="pralg_2.html#E7:38_1_1_3_1_3_1_2"><i><font color="Green" title="E34">A20</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:38_1_1_3_1_3_1"/><b>then </b><i><font color="Green" title="E31">A23</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">I</font>,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="pralg_2.html#K8" title="PRALG_2:func.8">|.</a><span class="default"><font color="Maroon" title="c11">A'</font></span><a href="pralg_2.html#K8" title="PRALG_2:func.8">.|</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_1"><i><font color="Green" title="E30">A16</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/><a NAME="E7:38_1_1_3_1_3_1"/><b>then </b>
 <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">I</font>,<span class="p2"><a href="pralg_2.html#K8" title="PRALG_2:func.8">|.</a><span class="default"><font color="Maroon" title="c11">A'</font></span><a href="pralg_2.html#K8" title="PRALG_2:func.8">.|</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_6.html#T85" title="FUNCT_6:th.85">FUNCT_6:85</a></i>;<br/><a NAME="E8:38_1_1_3_1_3_1"/><b>then </b><i><font color="Green" title="E32">A24</font></i>: 
 ex <font color="Olive" title="b1">h</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">I</font>,<span class="p1"><a href="pralg_2.html#K8" title="PRALG_2:func.8">|.</a><span class="default"><font color="Maroon" title="c11">A'</font></span><a href="pralg_2.html#K8" title="PRALG_2:func.8">.|</a></span> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/><a NAME="E9:38_1_1_3_1_3_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">o</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3_1_3_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:38_1_1_3_1_3_1_3"/><i><font color="Green" title="E33">A25</font></i>: 
<font color="Maroon" title="c22">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c23">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:38_1_1_3_1_3_1_3"/><i><font color="Green" title="E34">A26</font></i>: 
( <font color="Maroon" title="c23">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">h</font> = <font color="Maroon" title="c22">j</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_1_3"><i><font color="Green" title="E34">A26</font></i></a></i>;<br/>
<a NAME="E5:38_1_1_3_1_3_1_3"/>
 ex <font color="Olive" title="b1">k</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">h</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="pralg_2.html#K8" title="PRALG_2:func.8">|.</a><span class="default"><font color="Maroon" title="c11">A'</font></span><a href="pralg_2.html#K8" title="PRALG_2:func.8">.|</a></span> )
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1_3_1"><i><font color="Green" title="E32">A24</font></i></a>, <a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_1_3"><i><font color="Green" title="E33">A25</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E6:38_1_1_3_1_3_1_3"/><b>then </b><i><font color="Green" title="E35">A27</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c24">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E7:38_1_1_3_1_3_1_3"/>
for <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c24">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3_1_3_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c25">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_1_1_3_1_3_1_3_1"/>
<font color="Maroon" title="c25">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c26">i</font> = <font color="Maroon" title="c25">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">I'</font> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:38_1_1_3_1_3_1_3_1"/><i><font color="Green" title="E36">A28</font></i>: 
 ex <font color="Olive" title="b1">U0</font> being   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b1">U0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Olive" title="b1">U0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c13">o</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="pralg_2.html#D16" target="_self" title="PRALG_2:def.16">Def16</a></i>;<br/>
<a NAME="E4:38_1_1_3_1_3_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c13">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c26">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#T14" target="_self" title="PRALG_2:th.14">Th14</a></i>;<br/>
<a NAME="E5:38_1_1_3_1_3_1_3_1"/><b>then </b><i><font color="Green" title="E37">A29</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c13">o</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c26">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E6:38_1_1_3_1_3_1"><i><font color="Green" title="E31">A23</font></i></a>, <a class="txt" href="pralg_2.html#E8:38_1_1_3_1_3_1"><i><font color="Green" title="E32">A24</font></i></a>, <a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_1_3"><i><font color="Green" title="E34">A26</font></i></a>, <a class="ref" href="funct_6.html#T86" title="FUNCT_6:th.86">FUNCT_6:86</a></i>;<br/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c13">o</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c26">i</font></span>)</span></span>)</span> = 
 <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c13">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c26">i</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_1"><i><font color="Green" title="E27">A13</font></i></a>, <a class="ref" href="pralg_2.html#T11" target="_self" title="PRALG_2:th.11">Th11</a></i>
;<br/>
<a NAME="E7:38_1_1_3_1_3_1_3_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c13">o</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c26">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c13">o</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c26">i</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c13">o</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c26">i</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <font color="Maroon" title="c13">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c26">i</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1_3_1"><i><font color="Green" title="E32">A24</font></i></a>, <a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_1_3"><i><font color="Green" title="E34">A26</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E8:38_1_1_3_1_3_1_3_1"/><b>then </b>
<font color="Maroon" title="c24">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <font color="Maroon" title="c13">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c26">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_1_3_1"><i><font color="Green" title="E37">A29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:38_1_1_3_1_3_1_3_1"/>
<font color="Maroon" title="c24">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">b</font>
 <b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_1_3_1"><i><font color="Green" title="E36">A28</font></i></a>, <a class="ref" href="pralg_2.html#T10" target="_self" title="PRALG_2:th.10">Th10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:38_1_1_3_1_3_1_3"/>
<font color="Maroon" title="c22">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">o</font>
 <b>by </b><i><a class="txt" href="pralg_2.html#E11:38_1_1_3_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="txt" href="pralg_2.html#E6:38_1_1_3_1_3_1_3"><i><font color="Green" title="E35">A27</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E10:38_1_1_3_1_3_1"/>
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font>
 <b>by </b><i><a class="txt" href="pralg_2.html#E2:38_1_1_3_1_3_1"><i><font color="Green" title="E28">A14</font></i></a>, <a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_1"><i><font color="Green" title="E29">A15</font></i></a>, <a class="txt" href="pralg_2.html#E8:38_1_1_3_1_3_1"><i><font color="Green" title="E32">A24</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_3_1_3_2"><b>suppose </b></a><a NAME="E1:38_1_1_3_1_3_2"/><i><font color="Green" title="E27">A30</font></i>: 
 <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><i><font color="Green" title="E28">A31</font></i>: <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> = 
 <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">IFEQ</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_2.html#K2" title="PRALG_2:func.2">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" title="PRALG_2:func.3">Frege</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="pralg_2.html#E1:38_1_1_3_1"><i><font color="Green" title="E20">A6</font></i></a></i>
<br/>.= 
 <a href="pralg_2.html#K2" title="PRALG_2:func.2">Commute</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3" title="PRALG_2:func.3">Frege</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_2"><i><font color="Green" title="E27">A30</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
;<br/><a NAME="E3:38_1_1_3_1_3_2"/><i><font color="Green" title="E29">A32</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_2"><i><font color="Green" title="E27">A30</font></i></a>, <a class="ref" href="finseq_1.html#T26" title="FINSEQ_1:th.26">FINSEQ_1:26</a></i>;<br/><a NAME="E4:38_1_1_3_1_3_2"/><i><font color="Green" title="E30">A33</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K2" title="PRALG_2:func.2">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" title="PRALG_2:func.3">Frege</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">o</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3_1_3_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:38_1_1_3_1_3_2_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K2" title="PRALG_2:func.2">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" title="PRALG_2:func.3">Frege</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">o</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3_1_3_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:38_1_1_3_1_3_2_2_1"/>
<font color="Maroon" title="c21">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K2" title="PRALG_2:func.2">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" title="PRALG_2:func.3">Frege</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c22">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:38_1_1_3_1_3_2_2_1"/><i><font color="Green" title="E31">A34</font></i>: 
( <font color="Maroon" title="c22">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3" title="PRALG_2:func.3">Frege</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c21">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c22">g</font> )
 <b>by </b><i><a class="ref" href="pralg_2.html#D6" target="_self" title="PRALG_2:def.6">Def6</a></i>;<br/>
<a NAME="E4:38_1_1_3_1_3_2_2_1"/><i><font color="Green" title="E32">A35</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">I'</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">I'</font> holds  <font color="Maroon" title="c22">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c13">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c22">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>,<span class="p1"><a href="pralg_2.html#K8" title="PRALG_2:func.8">|.</a><span class="default"><font color="Maroon" title="c11">A'</font></span><a href="pralg_2.html#K8" title="PRALG_2:func.8">.|</a></span> )
 
<b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2_2_1"><i><font color="Green" title="E31">A34</font></i></a>, <a class="ref" href="pralg_2.html#T17" target="_self" title="PRALG_2:th.17">Th17</a></i>;<br/>
<a NAME="E5:38_1_1_3_1_3_2_2_1"/><b>then </b><i><font color="Green" title="E33">A36</font></i>: 
<font color="Maroon" title="c22">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">I</font>,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>,<span class="p2"><a href="pralg_2.html#K8" title="PRALG_2:func.8">|.</a><span class="default"><font color="Maroon" title="c11">A'</font></span><a href="pralg_2.html#K8" title="PRALG_2:func.8">.|</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E6:38_1_1_3_1_3_2_2_1"/><b>then </b>
 <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">I</font>,<span class="p2"><a href="pralg_2.html#K8" title="PRALG_2:func.8">|.</a><span class="default"><font color="Maroon" title="c11">A'</font></span><a href="pralg_2.html#K8" title="PRALG_2:func.8">.|</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2"><i><font color="Green" title="E29">A32</font></i></a>, <a class="ref" href="funct_6.html#T85" title="FUNCT_6:th.85">FUNCT_6:85</a></i>;<br/>
<a NAME="E7:38_1_1_3_1_3_2_2_1"/><b>then </b><i><font color="Green" title="E34">A37</font></i>: 
 ex <font color="Olive" title="b1">h</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c22">g</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">I</font>,<span class="p1"><a href="pralg_2.html#K8" title="PRALG_2:func.8">|.</a><span class="default"><font color="Maroon" title="c11">A'</font></span><a href="pralg_2.html#K8" title="PRALG_2:func.8">.|</a></span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<b>set </b><font color="Maroon" title="c23">cg</font> =  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c22">g</font>;<br/>
<a NAME="E8:38_1_1_3_1_3_2_2_1"/>
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c22">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3_1_3_2_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_1_1_3_1_3_2_2_1_1"/><i><font color="Green" title="E35">A38</font></i>: 
<font color="Maroon" title="c24">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:38_1_1_3_1_3_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c25">s</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">y</font> as   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="pralg_2.html#E6:38_1_1_3_1"><i><font color="Green" title="E23">A9</font></i></a></i>;<br/>
<i><font color="Green" title="E36">A39</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">y</font> = 
<span class="p1">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">s</font>
<b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_2_2_1_1"><i><font color="Green" title="E35">A38</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c25">s</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#D17" target="_self" title="PRALG_2:def.17">Def17</a></i>
;<br/>
<a NAME="E5:38_1_1_3_1_3_2_2_1_1"/><i><font color="Green" title="E37">A40</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c25">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E6:38_1_1_3_1_3_2_2_1_1"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c22">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c22">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="pralg_2.html#E7:38_1_1_3_1_3_2_2_1"><i><font color="Green" title="E34">A37</font></i></a>, <a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_2_2_1_1"><i><font color="Green" title="E35">A38</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c26">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E8:38_1_1_3_1_3_2_2_1_1"/><i><font color="Green" title="E38">A41</font></i>: 
( <font color="Maroon" title="c26">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c22">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">y</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c26">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c26">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="pralg_2.html#K8" title="PRALG_2:func.8">|.</a><span class="default"><font color="Maroon" title="c11">A'</font></span><a href="pralg_2.html#K8" title="PRALG_2:func.8">.|</a></span> )
 <b>by </b><i><a class="txt" href="pralg_2.html#E7:38_1_1_3_1_3_2_2_1"><i><font color="Green" title="E34">A37</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E9:38_1_1_3_1_3_2_2_1_1"/>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c25">s</font></span>)</span> holds <br/><font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c25">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3_1_3_2_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c27">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_1_1_3_1_3_2_2_1_1_2"/>
<font color="Maroon" title="c27">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c25">s</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c28">i</font> = <font color="Maroon" title="c27">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">I'</font> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:38_1_1_3_1_3_2_2_1_1_2"/><i><font color="Green" title="E39">A42</font></i>: 
 ex <font color="Olive" title="b1">U0</font> being   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b1">U0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">i</font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c25">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Olive" title="b1">U0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">s</font> )
 
<b>by </b><i><a class="ref" href="pralg_2.html#D16" target="_self" title="PRALG_2:def.16">Def16</a></i>;<br/>
<a NAME="E4:38_1_1_3_1_3_2_2_1_1_2"/><i><font color="Green" title="E40">A43</font></i>: 
( <font color="Maroon" title="c22">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c13">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c28">i</font></span>)</span> &amp; <font color="Maroon" title="c22">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c22">g</font> )
 
<b>by </b><i><a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_2_1"><i><font color="Green" title="E32">A35</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c29">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E6:38_1_1_3_1_3_2_2_1_1_2"/><i><font color="Green" title="E41">A44</font></i>: 
( <font color="Maroon" title="c29">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">i</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c29">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="pralg_2.html#K8" title="PRALG_2:func.8">|.</a><span class="default"><font color="Maroon" title="c11">A'</font></span><a href="pralg_2.html#K8" title="PRALG_2:func.8">.|</a></span> )
 <b>by </b><i><a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_2_1"><i><font color="Green" title="E32">A35</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E7:38_1_1_3_1_3_2_2_1_1_2"/><i><font color="Green" title="E42">A45</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<i><font color="Green" title="E43">A46</font></i>:  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c13">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c28">i</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">o</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D9" title="MSUALG_1:def.9">MSUALG_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_3_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="pralg_2.html#E7:38_1_1_3_1_3_2_2_1_1_2"><i><font color="Green" title="E42">A45</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_1.html#D6" title="MSUALG_1:def.6">MSUALG_1:def 6</a></i>
<br/>.= 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D19" title="PBOOLE:def.19">PBOOLE:def 19</a></i>
;<br/>
<a NAME="E9:38_1_1_3_1_3_2_2_1_1_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E10:38_1_1_3_1_3_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E44">A47</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E6:38_1_1_3_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E11:38_1_1_3_1_3_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E45">A48</font></i>: 
<font color="Maroon" title="c29">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">y</font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_2_2_1_1"><i><font color="Green" title="E35">A38</font></i></a>, <a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_2_1_1_2"><i><font color="Green" title="E40">A43</font></i></a>, <a class="txt" href="pralg_2.html#E6:38_1_1_3_1_3_2_2_1_1_2"><i><font color="Green" title="E41">A44</font></i></a>, <a class="txt" href="pralg_2.html#E8:38_1_1_3_1_3_2_2_1_1_2"><i><font color="Green" title="E43">A46</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E12:38_1_1_3_1_3_2_2_1_1_2"/>
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c28">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">s</font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_2_2_1_1"><i><font color="Green" title="E35">A38</font></i></a>, <a class="txt" href="pralg_2.html#E10:38_1_1_3_1_3_2_2_1_1_2"><i><font color="Green" title="E44">A47</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<b>hence </b><a NAME="E13:38_1_1_3_1_3_2_2_1_1_2"/>
<font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c25">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">z</font>
 <b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_2_1"><i><font color="Green" title="E33">A36</font></i></a>, <a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_2_2_1_1"><i><font color="Green" title="E35">A38</font></i></a>, <a class="txt" href="pralg_2.html#E8:38_1_1_3_1_3_2_2_1_1"><i><font color="Green" title="E38">A41</font></i></a>, <a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2_2_1_1_2"><i><font color="Green" title="E39">A42</font></i></a>, <a class="txt" href="pralg_2.html#E6:38_1_1_3_1_3_2_2_1_1_2"><i><font color="Green" title="E41">A44</font></i></a>, <a class="txt" href="pralg_2.html#E11:38_1_1_3_1_3_2_2_1_1_2"><i><font color="Green" title="E45">A48</font></i></a>, <a class="ref" href="funct_6.html#T86" title="FUNCT_6:th.86">FUNCT_6:86</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:38_1_1_3_1_3_2_2_1_1"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c22">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">y</font>
 <b>by </b><i><a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_2_1_1"><i><font color="Green" title="E36">A39</font></i></a>, <a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_2_1_1"><i><font color="Green" title="E37">A40</font></i></a>, <a class="txt" href="pralg_2.html#E8:38_1_1_3_1_3_2_2_1_1"><i><font color="Green" title="E38">A41</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:38_1_1_3_1_3_2_2_1"/>
<font color="Maroon" title="c21">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">o</font>
 <b>by </b><i><a class="txt" href="pralg_2.html#E5:38_1_1_3_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="pralg_2.html#E8:38_1_1_3_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2_2_1"><i><font color="Green" title="E31">A34</font></i></a>, <a class="txt" href="pralg_2.html#E7:38_1_1_3_1_3_2_2_1"><i><font color="Green" title="E34">A37</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c21">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:38_1_1_3_1_3_2_2"/>
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">o</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c22">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:38_1_1_3_1_3_2_2"/><i><font color="Green" title="E31">A49</font></i>: 
( <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c22">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )
 <b>by </b><i><a class="txt" href="pralg_2.html#E5:38_1_1_3_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<a NAME="E5:38_1_1_3_1_3_2_2"/><i><font color="Green" title="E32">A50</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c22">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">I</font>,<span class="p1"><a href="pralg_2.html#K8" title="PRALG_2:func.8">|.</a><span class="default"><font color="Maroon" title="c11">A'</font></span><a href="pralg_2.html#K8" title="PRALG_2:func.8">.|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3_1_3_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:38_1_1_3_1_3_2_2_2"/>
<font color="Maroon" title="c23">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c22">g</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c24">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:38_1_1_3_1_3_2_2_2"/><i><font color="Green" title="E33">A51</font></i>: 
( <font color="Maroon" title="c24">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c22">g</font> &amp; <font color="Maroon" title="c22">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">a</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:38_1_1_3_1_3_2_2_2"/><i><font color="Green" title="E34">A52</font></i>: 
<font color="Maroon" title="c23">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">b</font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_2"><i><font color="Green" title="E31">A49</font></i></a>, <a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2_2_2"><i><font color="Green" title="E33">A51</font></i></a></i>;<br/>
<a NAME="E5:38_1_1_3_1_3_2_2_2"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_2"><i><font color="Green" title="E31">A49</font></i></a>, <a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2_2_2"><i><font color="Green" title="E33">A51</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c25">cr</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">b</font> as   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="pralg_2.html#E6:38_1_1_3_1"><i><font color="Green" title="E23">A9</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">b</font> = 
<span class="p1">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">cr</font>
<b>by </b><i><a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_2"><i><font color="Green" title="E31">A49</font></i></a>, <a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2_2_2"><i><font color="Green" title="E33">A51</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c25">cr</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#D17" target="_self" title="PRALG_2:def.17">Def17</a></i>
;<br/>
<b>then consider </b><font color="Maroon" title="c26">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E9:38_1_1_3_1_3_2_2_2"/><i><font color="Green" title="E35">A53</font></i>: 
( <font color="Maroon" title="c26">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">a</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c26">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c25">cr</font></span>)</span> &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c25">cr</font></span>)</span> holds <br/><font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c25">cr</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> ) )
 <b>by </b><i><a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_2_2"><i><font color="Green" title="E34">A52</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<a NAME="E10:38_1_1_3_1_3_2_2_2"/><i><font color="Green" title="E36">A54</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c25">cr</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E11:38_1_1_3_1_3_2_2_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c26">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="pralg_2.html#K8" title="PRALG_2:func.8">|.</a><span class="default"><font color="Maroon" title="c11">A'</font></span><a href="pralg_2.html#K8" title="PRALG_2:func.8">.|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3_1_3_2_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c27">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:38_1_1_3_1_3_2_2_2_2"/>
<font color="Maroon" title="c27">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c26">h</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c28">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:38_1_1_3_1_3_2_2_2_2"/><i><font color="Green" title="E37">A55</font></i>: 
( <font color="Maroon" title="c28">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c26">h</font> &amp; <font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">j</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c29">i</font> = <font color="Maroon" title="c28">c</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">I'</font> <b>by </b><i><a class="txt" href="pralg_2.html#E9:38_1_1_3_1_3_2_2_2"><i><font color="Green" title="E35">A53</font></i></a>, <a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2_2_2_2"><i><font color="Green" title="E37">A55</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:38_1_1_3_1_3_2_2_2_2"/><i><font color="Green" title="E38">A56</font></i>: 
<font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c25">cr</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E9:38_1_1_3_1_3_2_2_2"><i><font color="Green" title="E35">A53</font></i></a>, <a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2_2_2_2"><i><font color="Green" title="E37">A55</font></i></a></i>;<br/>
<a NAME="E6:38_1_1_3_1_3_2_2_2_2"/><i><font color="Green" title="E39">A57</font></i>: 
 ex <font color="Olive" title="b1">U0</font> being   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b1">U0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c25">cr</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Olive" title="b1">U0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">cr</font> )
 
<b>by </b><i><a class="ref" href="pralg_2.html#D16" target="_self" title="PRALG_2:def.16">Def16</a></i>;<br/>
<a NAME="E7:38_1_1_3_1_3_2_2_2_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c29">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E8:38_1_1_3_1_3_2_2_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c25">cr</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c29">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E6:38_1_1_3_1_3_2_2_2_2"><i><font color="Green" title="E39">A57</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:38_1_1_3_1_3_2_2_2_2"/><b>then </b><i><font color="Green" title="E40">A58</font></i>: 
<font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="pralg_2.html#K7" title="PRALG_2:func.7">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c29">i</font></span>)</span></span><a href="pralg_2.html#K7" title="PRALG_2:func.7">.|</a></span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_2_2_2"><i><font color="Green" title="E38">A56</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E10:38_1_1_3_1_3_2_2_2_2"/>
<span class="p1"><a href="pralg_2.html#K7" title="PRALG_2:func.7">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c29">i</font></span>)</span></span><a href="pralg_2.html#K7" title="PRALG_2:func.7">.|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="pralg_2.html#K7" title="PRALG_2:func.7">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b1">d</font></span>)</span></span><a href="pralg_2.html#K7" title="PRALG_2:func.7">.|</a></span> where <font color="Olive" title="b1">d</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">I'</font> : verum </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E11:38_1_1_3_1_3_2_2_2_2"/>
<font color="Maroon" title="c27">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="pralg_2.html#K8" title="PRALG_2:func.8">|.</a><span class="default"><font color="Maroon" title="c11">A'</font></span><a href="pralg_2.html#K8" title="PRALG_2:func.8">.|</a></span>
 <b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2_2_2_2"><i><font color="Green" title="E37">A55</font></i></a>, <a class="txt" href="pralg_2.html#E9:38_1_1_3_1_3_2_2_2_2"><i><font color="Green" title="E40">A58</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:38_1_1_3_1_3_2_2_2"/>
<font color="Maroon" title="c23">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">I</font>,<span class="p1"><a href="pralg_2.html#K8" title="PRALG_2:func.8">|.</a><span class="default"><font color="Maroon" title="c11">A'</font></span><a href="pralg_2.html#K8" title="PRALG_2:func.8">.|</a></span>
 <b>by </b><i><a class="txt" href="pralg_2.html#E9:38_1_1_3_1_3_2_2_2"><i><font color="Green" title="E35">A53</font></i></a>, <a class="txt" href="pralg_2.html#E10:38_1_1_3_1_3_2_2_2"><i><font color="Green" title="E36">A54</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:38_1_1_3_1_3_2_2"/><b>then </b><i><font color="Green" title="E33">A59</font></i>: 
<font color="Maroon" title="c22">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">I</font>,<span class="p2"><a href="pralg_2.html#K8" title="PRALG_2:func.8">|.</a><span class="default"><font color="Maroon" title="c11">A'</font></span><a href="pralg_2.html#K8" title="PRALG_2:func.8">.|</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_2"><i><font color="Green" title="E31">A49</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E7:38_1_1_3_1_3_2_2"/><b>then </b>
 <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">I</font>,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>,<span class="p2"><a href="pralg_2.html#K8" title="PRALG_2:func.8">|.</a><span class="default"><font color="Maroon" title="c11">A'</font></span><a href="pralg_2.html#K8" title="PRALG_2:func.8">.|</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2"><i><font color="Green" title="E29">A32</font></i></a>, <a class="ref" href="funct_6.html#T85" title="FUNCT_6:th.85">FUNCT_6:85</a></i>;<br/>
<a NAME="E8:38_1_1_3_1_3_2_2"/><b>then </b><i><font color="Green" title="E34">A60</font></i>: 
 ex <font color="Olive" title="b1">h</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c22">g</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>,<span class="p1"><a href="pralg_2.html#K8" title="PRALG_2:func.8">|.</a><span class="default"><font color="Maroon" title="c11">A'</font></span><a href="pralg_2.html#K8" title="PRALG_2:func.8">.|</a></span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E9:38_1_1_3_1_3_2_2"/><i><font color="Green" title="E35">A61</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3" title="PRALG_2:func.3">Frege</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E10:38_1_1_3_1_3_2_2"/><i><font color="Green" title="E36">A62</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">I'</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">I'</font> holds  <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c13">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="pralg_2.html#T18" target="_self" title="PRALG_2:th.18">Th18</a></i>;<br/>
<a NAME="E11:38_1_1_3_1_3_2_2"/>
for <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c22">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3_1_3_2_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_1_1_3_1_3_2_2_3"/>
<font color="Maroon" title="c23">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c24">ii</font> = <font color="Maroon" title="c23">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">I'</font> <b>by </b><i><a class="ref" href="pralg_2.html#T18" target="_self" title="PRALG_2:th.18">Th18</a></i>;<br/>
<a NAME="E3:38_1_1_3_1_3_2_2_3"/><i><font color="Green" title="E37">A63</font></i>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">ii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c13">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">ii</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#T18" target="_self" title="PRALG_2:th.18">Th18</a></i>;<br/>
<b>set </b><font color="Maroon" title="c25">cg</font> =  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c22">g</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c26">h</font> = <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c22">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">ii</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<a NAME="E5:38_1_1_3_1_3_2_2_3"/><i><font color="Green" title="E38">A64</font></i>: 
(  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c13">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">ii</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">ii</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">ii</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="pralg_2.html#T10" target="_self" title="PRALG_2:th.10">Th10</a></i>;<br/>
<a NAME="E6:38_1_1_3_1_3_2_2_3"/>
<font color="Maroon" title="c26">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c22">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1_3_2_2"><i><font color="Green" title="E34">A60</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:38_1_1_3_1_3_2_2_3"/><b>then </b><i><font color="Green" title="E39">A65</font></i>: 
 ex <font color="Olive" title="b1">s</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">h</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="pralg_2.html#K8" title="PRALG_2:func.8">|.</a><span class="default"><font color="Maroon" title="c11">A'</font></span><a href="pralg_2.html#K8" title="PRALG_2:func.8">.|</a></span> )
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1_3_2_2"><i><font color="Green" title="E34">A60</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<b>set </b><font color="Maroon" title="c27">So</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">ii</font></span>)</span>;<br/>
<a NAME="E8:38_1_1_3_1_3_2_2_3"/>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">ii</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">ii</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3_1_3_2_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c28">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_1_1_3_1_3_2_2_3_1"/><i><font color="Green" title="E40">A66</font></i>: 
<font color="Maroon" title="c28">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">ii</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:38_1_1_3_1_3_2_2_3_1"/><b>then </b>
<font color="Maroon" title="c22">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c22">g</font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_2"><i><font color="Green" title="E31">A49</font></i></a>, <a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_2_3"><i><font color="Green" title="E38">A64</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c29">k</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:38_1_1_3_1_3_2_2_3_1"/><i><font color="Green" title="E41">A67</font></i>: 
( <font color="Maroon" title="c29">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">a</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c29">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="pralg_2.html#K8" title="PRALG_2:func.8">|.</a><span class="default"><font color="Maroon" title="c11">A'</font></span><a href="pralg_2.html#K8" title="PRALG_2:func.8">.|</a></span> )
 <b>by </b><i><a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_2"><i><font color="Green" title="E32">A50</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E5:38_1_1_3_1_3_2_2_3_1"/><i><font color="Green" title="E42">A68</font></i>: 
<font color="Maroon" title="c29">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">ii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">a</font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E6:38_1_1_3_1_3_2_2"><i><font color="Green" title="E33">A59</font></i></a>, <a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_2_3"><i><font color="Green" title="E38">A64</font></i></a>, <a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_2_2_3_1"><i><font color="Green" title="E40">A66</font></i></a>, <a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_2_3_1"><i><font color="Green" title="E41">A67</font></i></a>, <a class="ref" href="funct_6.html#T86" title="FUNCT_6:th.86">FUNCT_6:86</a></i>;<br/>
<a NAME="E6:38_1_1_3_1_3_2_2_3_1"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_2_3"><i><font color="Green" title="E38">A64</font></i></a>, <a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_2_2_3_1"><i><font color="Green" title="E40">A66</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c30">s</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">a</font> as   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="pralg_2.html#E6:38_1_1_3_1"><i><font color="Green" title="E23">A9</font></i></a></i>;<br/>
<a NAME="E8:38_1_1_3_1_3_2_2_3_1"/><i><font color="Green" title="E43">A69</font></i>: 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">ii</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">ii</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">s</font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_2_2_3_1"><i><font color="Green" title="E40">A66</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<a NAME="E9:38_1_1_3_1_3_2_2_3_1"/><i><font color="Green" title="E44">A70</font></i>: 
<font color="Maroon" title="c29">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">a</font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_2"><i><font color="Green" title="E31">A49</font></i></a>, <a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_2_3"><i><font color="Green" title="E38">A64</font></i></a>, <a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_2_2_3_1"><i><font color="Green" title="E40">A66</font></i></a>, <a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_2_3_1"><i><font color="Green" title="E41">A67</font></i></a></i>;<br/>
<i><font color="Green" title="E45">A71</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">a</font> = 
<span class="p1">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">s</font>
<b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_2_3"><i><font color="Green" title="E38">A64</font></i></a>, <a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_2_2_3_1"><i><font color="Green" title="E40">A66</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c30">s</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#D17" target="_self" title="PRALG_2:def.17">Def17</a></i>
;<br/>
<a NAME="E11:38_1_1_3_1_3_2_2_3_1"/><i><font color="Green" title="E46">A72</font></i>: 
 ex <font color="Olive" title="b1">U0</font> being   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b1">U0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">ii</font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c30">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">ii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Olive" title="b1">U0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">s</font> )
 
<b>by </b><i><a class="ref" href="pralg_2.html#D16" target="_self" title="PRALG_2:def.16">Def16</a></i>;<br/>
<a NAME="E12:38_1_1_3_1_3_2_2_3_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c30">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>hence </b><a NAME="E13:38_1_1_3_1_3_2_2_3_1"/>
<font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c24">ii</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">a</font>
 <b>by </b><i><a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_2_3_1"><i><font color="Green" title="E42">A68</font></i></a>, <a class="txt" href="pralg_2.html#E8:38_1_1_3_1_3_2_2_3_1"><i><font color="Green" title="E43">A69</font></i></a>, <a class="txt" href="pralg_2.html#E9:38_1_1_3_1_3_2_2_3_1"><i><font color="Green" title="E44">A70</font></i></a>, <a class="txt" href="pralg_2.html#E10:38_1_1_3_1_3_2_2_3_1"><i><font color="Green" title="E45">A71</font></i></a>, <a class="txt" href="pralg_2.html#E11:38_1_1_3_1_3_2_2_3_1"><i><font color="Green" title="E46">A72</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:38_1_1_3_1_3_2_2_3"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c22">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">j</font>
 <b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2_2_3"><i><font color="Green" title="E37">A63</font></i></a>, <a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_2_3"><i><font color="Green" title="E38">A64</font></i></a>, <a class="txt" href="pralg_2.html#E7:38_1_1_3_1_3_2_2_3"><i><font color="Green" title="E39">A65</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:38_1_1_3_1_3_2_2"/><b>then </b><i><font color="Green" title="E37">A73</font></i>: 
 <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3" title="PRALG_2:func.3">Frege</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E8:38_1_1_3_1_3_2_2"><i><font color="Green" title="E34">A60</font></i></a>, <a class="txt" href="pralg_2.html#E9:38_1_1_3_1_3_2_2"><i><font color="Green" title="E35">A61</font></i></a>, <a class="txt" href="pralg_2.html#E10:38_1_1_3_1_3_2_2"><i><font color="Green" title="E36">A62</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E13:38_1_1_3_1_3_2_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_2"><i><font color="Green" title="E27">A30</font></i></a>, <a class="ref" href="finseq_1.html#T26" title="FINSEQ_1:th.26">FINSEQ_1:26</a></i>;<br/>
<a NAME="E14:38_1_1_3_1_3_2_2"/><b>then </b>
 <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c22">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g</font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E6:38_1_1_3_1_3_2_2"><i><font color="Green" title="E33">A59</font></i></a>, <a class="ref" href="funct_6.html#T87" title="FUNCT_6:th.87">FUNCT_6:87</a></i>;<br/>
<b>hence </b><a NAME="E15:38_1_1_3_1_3_2_2"/>
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K2" title="PRALG_2:func.2">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" title="PRALG_2:func.3">Frege</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_2"><i><font color="Green" title="E31">A49</font></i></a>, <a class="txt" href="pralg_2.html#E12:38_1_1_3_1_3_2_2"><i><font color="Green" title="E37">A73</font></i></a>, <a class="ref" href="pralg_2.html#D6" target="_self" title="PRALG_2:def.6">Def6</a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon" title="c21">rs</font> =  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c13">o</font>;<br/><b>set </b><font color="Maroon" title="c22">CA</font> =  <a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c13">o</font></span>)</span>;<br/><a NAME="E5:38_1_1_3_1_3_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K2" title="PRALG_2:func.2">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" title="PRALG_2:func.3">Frege</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">o</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3_1_3_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:38_1_1_3_1_3_2_3"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K2" title="PRALG_2:func.2">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" title="PRALG_2:func.3">Frege</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c24">g</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:38_1_1_3_1_3_2_3"/><i><font color="Green" title="E31">A74</font></i>: 
( <font color="Maroon" title="c24">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K2" title="PRALG_2:func.2">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" title="PRALG_2:func.3">Frege</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K2" title="PRALG_2:func.2">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" title="PRALG_2:func.3">Frege</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c25">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:38_1_1_3_1_3_2_3"/><i><font color="Green" title="E32">A75</font></i>: 
( <font color="Maroon" title="c25">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3" title="PRALG_2:func.3">Frege</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c24">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c25">f</font> )
 <b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2_3"><i><font color="Green" title="E31">A74</font></i></a>, <a class="ref" href="pralg_2.html#D6" target="_self" title="PRALG_2:def.6">Def6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c26">g</font> = <font color="Maroon" title="c24">g</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_3"><i><font color="Green" title="E32">A75</font></i></a></i>;<br/>
<a NAME="E7:38_1_1_3_1_3_2_3"/>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c25">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">I'</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">I'</font> holds  <font color="Maroon" title="c25">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c13">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c25">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>,<span class="p1"><a href="pralg_2.html#K8" title="PRALG_2:func.8">|.</a><span class="default"><font color="Maroon" title="c11">A'</font></span><a href="pralg_2.html#K8" title="PRALG_2:func.8">.|</a></span> )
 
<b>by </b><i><a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_3"><i><font color="Green" title="E32">A75</font></i></a>, <a class="ref" href="pralg_2.html#T17" target="_self" title="PRALG_2:th.17">Th17</a></i>;<br/>
<a NAME="E8:38_1_1_3_1_3_2_3"/><b>then </b><i><font color="Green" title="E33">A76</font></i>: 
<font color="Maroon" title="c25">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">I</font>,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>,<span class="p2"><a href="pralg_2.html#K8" title="PRALG_2:func.8">|.</a><span class="default"><font color="Maroon" title="c11">A'</font></span><a href="pralg_2.html#K8" title="PRALG_2:func.8">.|</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E9:38_1_1_3_1_3_2_3"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_3_1_3_2"><i><font color="Green" title="E27">A30</font></i></a>, <a class="ref" href="finseq_1.html#T26" title="FINSEQ_1:th.26">FINSEQ_1:26</a></i>;<br/>
<a NAME="E10:38_1_1_3_1_3_2_3"/><b>then </b>
 <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c26">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">f</font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_3"><i><font color="Green" title="E32">A75</font></i></a>, <a class="txt" href="pralg_2.html#E8:38_1_1_3_1_3_2_3"><i><font color="Green" title="E33">A76</font></i></a>, <a class="ref" href="funct_6.html#T87" title="FUNCT_6:th.87">FUNCT_6:87</a></i>;<br/>
<a NAME="E11:38_1_1_3_1_3_2_3"/><b>then </b><i><font color="Green" title="E34">A77</font></i>: 
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3" title="PRALG_2:func.3">Frege</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">f</font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K3" title="PRALG_2:func.3">Frege</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3" title="PRALG_2:func.3">Frege</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2_3"><i><font color="Green" title="E31">A74</font></i></a>, <a class="txt" href="pralg_2.html#E5:38_1_1_3_1_3_2_3"><i><font color="Green" title="E32">A75</font></i></a>, <a class="ref" href="pralg_2.html#D6" target="_self" title="PRALG_2:def.6">Def6</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c27">h</font> = <font color="Maroon" title="c23">x</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<a NAME="E13:38_1_1_3_1_3_2_3"/><i><font color="Green" title="E35">A78</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c27">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">I'</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">I'</font> holds  <font color="Maroon" title="c27">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <font color="Maroon" title="c13">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="pralg_2.html#E11:38_1_1_3_1_3_2_3"><i><font color="Green" title="E34">A77</font></i></a>, <a class="ref" href="pralg_2.html#T16" target="_self" title="PRALG_2:th.16">Th16</a></i>;<br/>
<a NAME="E14:38_1_1_3_1_3_2_3"/><i><font color="Green" title="E36">A79</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E15:38_1_1_3_1_3_2_3"/>
for <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c27">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3_1_3_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c28">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_1_1_3_1_3_2_3_1"/>
<font color="Maroon" title="c28">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c29">i</font> = <font color="Maroon" title="c28">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">I'</font> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:38_1_1_3_1_3_2_3_1"/><i><font color="Green" title="E37">A80</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c29">i</font></span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E9:38_1_1_3_1"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:38_1_1_3_1_3_2_3_1"/><i><font color="Green" title="E38">A81</font></i>: 
 ex <font color="Olive" title="b1">U0</font> being   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b1">U0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c29">i</font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Olive" title="b1">U0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c13">o</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="pralg_2.html#D16" target="_self" title="PRALG_2:def.16">Def16</a></i>;<br/>
 <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <font color="Maroon" title="c13">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c29">i</font></span>)</span> = 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c29">i</font></span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">o</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D10" title="MSUALG_1:def.10">MSUALG_1:def 10</a></i>
<br/>.= 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c11">A'</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c29">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_2.html#E1:38_1_1_3_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="pralg_2.html#E9:38_1_1_3_1"><i><font color="Green" title="E25">A11</font></i></a>, <a class="txt" href="pralg_2.html#E3:38_1_1_3_1_3_2_3_1"><i><font color="Green" title="E37">A80</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font>
<b>by </b><i><a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2_3_1"><i><font color="Green" title="E38">A81</font></i></a>, <a class="ref" href="msualg_1.html#D7" title="MSUALG_1:def.7">MSUALG_1:def 7</a></i>
;<br/>
<b>hence </b><a NAME="E6:38_1_1_3_1_3_2_3_1"/>
<font color="Maroon" title="c27">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">j</font>
 <b>by </b><i><a class="txt" href="pralg_2.html#E11:38_1_1_3_1_3_2_3"><i><font color="Green" title="E34">A77</font></i></a>, <a class="ref" href="pralg_2.html#T16" target="_self" title="PRALG_2:th.16">Th16</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:38_1_1_3_1_3_2_3"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">o</font>
 <b>by </b><i><a class="txt" href="pralg_2.html#E11:38_1_1_3_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="txt" href="pralg_2.html#E13:38_1_1_3_1_3_2_3"><i><font color="Green" title="E35">A78</font></i></a>, <a class="txt" href="pralg_2.html#E14:38_1_1_3_1_3_2_3"><i><font color="Green" title="E36">A79</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:38_1_1_3_1_3_2"/>
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font>
 <b>by </b><i><a class="txt" href="pralg_2.html#E2:38_1_1_3_1_3_2"><i><font color="Green" title="E28">A31</font></i></a>, <a class="txt" href="pralg_2.html#E4:38_1_1_3_1_3_2"><i><font color="Green" title="E30">A33</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c12">f</font> = <font color="Maroon" title="c9">f</font> as    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c2">S</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/><b>take </b><font color="Maroon" title="c13">f</font> = <font color="Maroon" title="c12">f</font>;<br/><b>let </b><font color="Maroon" title="c14">o</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c2">S</font>;<br/><a NAME="E6:38_1_1_3"/>
the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">S</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="msualg_1.html#D5" title="MSUALG_1:def.5">MSUALG_1:def 5</a></i>;<br/><b>hence </b><a NAME="E7:38_1_1_3"/>
<font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K12" title="FUNCOP_1:func.12">IFEQ</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c14">o</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c14">o</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_2.html#K2" title="PRALG_2:func.2">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3" title="PRALG_2:func.3">Frege</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K13" title="PRALG_2:func.13">?.</a> <font color="Maroon" title="c14">o</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_2.html#E3:38_1_1"><i><font color="Green" title="E19">A2</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E8:38_1_1"/>
<font color="Maroon" title="c1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c10">f</font> being    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c2">S</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">S</font>;<br/>
<b>take </b>
<font color="Maroon" title="c10">f</font>
;<br/>


</div>
