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

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is    <a href="pralg_2.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span> : for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>  st <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Olive">i</font>,<font color="Olive">j</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">C</font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E2:14_1_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> holds  <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">s</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S5">PBOOLE:sch 5</a>();<br/></i>
<a NAME="E3:14_1_1"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> holds <br/><font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">OAF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:14_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">i</font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="pralg_2.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<font color="Maroon">s</font></span>)</span>] <b>means </b>for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>  st <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Olive">i</font>,<font color="Olive">j</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>;<br/>
<a NAME="E3:14_1_1_1"/><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">OAF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<font color="Maroon">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>;<br/>
<a NAME="E4:14_1_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is    <a href="pralg_2.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<font color="Maroon">s</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>] </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<font color="Maroon">s</font></span>)</span>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S10">FRAENKEL:sch 10</a>();<br/></i>
<b>hence </b><a NAME="E5:14_1_1_1"/>
<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">OAF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:14_1_1"/><b>then </b>
<font color="Maroon">C</font> <a href="pboole.html#R2">c=</a>  <a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">OAF</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">C</font> = <font color="Maroon">C</font> as    <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">OAF</font> <b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">C</font> as   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">OAF</font></span>)</span> <b>by </b><i><a class="ref" href="pralg_2.html#T20">PRALG_2:20</a></i>;<br/>
<a NAME="E7:14_1_1"/>
for <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font> holds  <font color="Maroon">C</font> <a href="msualg_2.html#R1">is_closed_on</a> <font color="Olive">o</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o</font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>;<br/>

<a NAME="E1:14_1_1_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p3">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">OAF</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">C</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:14_1_1_2_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p3">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">OAF</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">C</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">K</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">OAF</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:14_1_1_2_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">K</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:14_1_1_2_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p3">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">OAF</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E6:14_1_1_2_1"/><b>then </b><i><font color="Green">E45</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">OAF</font></span>)</span></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">MS</font> = <font color="Maroon">C</font> as    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E8:14_1_1_2_1"/><i><font color="Green">E46</font></i>: 
not the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D5">MSUALG_1:def 5</a></i>;<br/>
<a NAME="E9:14_1_1_2_1"/>
 <a href="relset_1.html#K4">dom</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:14_1_1_2_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E11:14_1_1_2_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a></i>;<br/>
<a NAME="E12:14_1_1_2_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">MS</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D19">PBOOLE:def 19</a></i>;<br/>
<a NAME="E13:14_1_1_2_1"/><i><font color="Green">E48</font></i>: 
 <a href="relset_1.html#K4">dom</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E14:14_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">OAF</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E15:14_1_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">OAF</font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D10">MSUALG_1:def 10</a></i>;<br/>
<a NAME="E16:14_1_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">OAF</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E17:14_1_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">OAF</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#T20">PRALG_2:20</a></i>;<br/>
<a NAME="E18:14_1_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">OAF</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>;<br/>
<a NAME="E19:14_1_1_2_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">x</font> is    <a href="pralg_2.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>;<br/><a NAME="E1:14_1_1_2_1_1"/>
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>  st <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Olive">i</font>,<font color="Olive">j</font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>;<br/><b>let </b><font color="Maroon">j</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>;<br/>



<b>assume </b><a NAME="E1:14_1_1_2_1_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font>
 ;<br/>

<i><font color="Green">E70</font></i>:  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">OAF</font></span>)</span> = 
the <a href="msualg_1.html#U5">Charact</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">OAF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_2.html#K14">OPS</a> <font color="Maroon">OAF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="pralg_2.html#T20">PRALG_2:20</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">OPE</font> = <span class="p1">(<span class="default"><a href="pralg_2.html#K14">OPS</a> <font color="Maroon">OAF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:14_1_1_2_1_1_1"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">MS</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> &amp; ( for <font color="Olive">t</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">MS</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">MS</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">t</font> ) )
 <b>by </b><i>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i/>;<br/>
<a NAME="E7:14_1_1_2_1_1_1"/><i><font color="Green">E73</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">MS</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E8:14_1_1_2_1_1_1"/><i><font color="Green">E74</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">MS</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2_1_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:14_1_1_2_1_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>
 ;<br/>

<a NAME="E2:14_1_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
;<br/>
<b>hence </b><a NAME="E3:14_1_1_2_1_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">MS</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:14_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<b>set </b><font color="Maroon">y1</font> = <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>;<br/>
<b>reconsider </b><font color="Maroon">Co</font> =  <a href="funct_6.html#K10">commute</a> <font color="Maroon">y</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E11:14_1_1_2_1_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font>,<span class="p2"><a href="pralg_2.html#K8">|.</a><span class="default"><font color="Maroon">OAF</font></span><a href="pralg_2.html#K8">.|</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2_1_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:14_1_1_2_1_1_1_3"/><i><font color="Green">E77</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E2:14_1_1_2_1_1_1_3"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font>,<span class="p1"><a href="pralg_2.html#K8">|.</a><span class="default"><font color="Maroon">OAF</font></span><a href="pralg_2.html#K8">.|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2_1_1_1_3_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:14_1_1_2_1_1_1_3_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">y</font>
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:14_1_1_2_1_1_1_3_1"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">y</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:14_1_1_2_1_1_1_3_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">MS</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E5:14_1_1_2_1_1_1_3_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">MS</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 
<b>by </b><i>, , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E6:14_1_1_2_1_1_1_3_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">MS</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E7:14_1_1_2_1_1_1_3_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E8:14_1_1_2_1_1_1_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Pa</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E10:14_1_1_2_1_1_1_3_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is    <a href="pralg_2.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<font color="Maroon">Pa</font></span>)</span> : for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>  st <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Olive">i</font>,<font color="Olive">j</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">Pa</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">z'</font> being    <a href="pralg_2.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<font color="Maroon">Pa</font></span>)</span><b> such that </b><br/><a NAME="E12:14_1_1_2_1_1_1_3_1"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; ( for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>  st <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Olive">i</font>,<font color="Olive">j</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">Pa</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">z'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> ) )
 ;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i/>;<br/>
<i><font color="Green">E82</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<font color="Maroon">Pa</font></span>)</span>
<b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<a NAME="E15:14_1_1_2_1_1_1_3_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="pralg_2.html#K8">|.</a><span class="default"><font color="Maroon">OAF</font></span><a href="pralg_2.html#K8">.|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2_1_1_1_3_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:14_1_1_2_1_1_1_3_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">z</font>
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:14_1_1_2_1_1_1_3_1_2"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font> &amp; <font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:14_1_1_2_1_1_1_3_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<font color="Maroon">Pa</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E5:14_1_1_2_1_1_1_3_1_2"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<font color="Maroon">Pa</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font>
 
<b>by </b><i>, , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r'</font> = <font color="Maroon">r</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font> <b>by </b><i><a class="txt" href="msalimit.html#E2:14_1_1"><i><font color="Green">E29</font></i></a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">r'</font> = <font color="Maroon">r'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font> ;<br/>
<a NAME="E8:14_1_1_2_1_1_1_3_1_2"/>
<font color="Maroon">r'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font>
 
;<br/>
<b>then consider </b><font color="Maroon">U0</font> being    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E10:14_1_1_2_1_1_1_3_1_2"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">U0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">OAF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<font color="Maroon">Pa</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Pa</font> )
 <b>by </b><i><a class="ref" href="pralg_2.html#D16">PRALG_2:def 16</a></i>;<br/>
<a NAME="E11:14_1_1_2_1_1_1_3_1_2"/>
 <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">r'</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E12:14_1_1_2_1_1_1_3_1_2"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">r'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">Pa</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">r'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E13:14_1_1_2_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">r'</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E14:14_1_1_2_1_1_1_3_1_2"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="pralg_2.html#K7">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">r'</font></span>)</span></span><a href="pralg_2.html#K7">.|</a></span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D13">PRALG_2:def 13</a></i>;<br/>
<a NAME="E15:14_1_1_2_1_1_1_3_1_2"/>
<span class="p1"><a href="pralg_2.html#K7">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">r'</font></span>)</span></span><a href="pralg_2.html#K7">.|</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="pralg_2.html#K7">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">k</font></span>)</span></span><a href="pralg_2.html#K7">.|</a></span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font> : verum </span>}</span> 
 
;<br/>
<a NAME="E16:14_1_1_2_1_1_1_3_1_2"/><b>then </b>
<span class="p1"><a href="pralg_2.html#K7">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">r'</font></span>)</span></span><a href="pralg_2.html#K7">.|</a></span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2"><a href="pralg_2.html#K7">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">k</font></span>)</span></span><a href="pralg_2.html#K7">.|</a></span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E17:14_1_1_2_1_1_1_3_1_2"/><b>then </b>
<span class="p1"><a href="pralg_2.html#K7">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">r'</font></span>)</span></span><a href="pralg_2.html#K7">.|</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="pralg_2.html#K8">|.</a><span class="default"><font color="Maroon">OAF</font></span><a href="pralg_2.html#K8">.|</a></span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D14">PRALG_2:def 14</a></i>;<br/>
<b>hence </b><a NAME="E18:14_1_1_2_1_1_1_3_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="pralg_2.html#K8">|.</a><span class="default"><font color="Maroon">OAF</font></span><a href="pralg_2.html#K8">.|</a></span>
 <b>by </b><i><a class="ref" href="msalimit.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:14_1_1_2_1_1_1_3_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font>,<span class="p1"><a href="pralg_2.html#K8">|.</a><span class="default"><font color="Maroon">OAF</font></span><a href="pralg_2.html#K8">.|</a></span>
 <b>by </b><i><a class="txt" href="msalimit.html#E2:14_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:14_1_1_2_1_1_1_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font>,<span class="p2"><a href="pralg_2.html#K8">|.</a><span class="default"><font color="Maroon">OAF</font></span><a href="pralg_2.html#K8">.|</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_2_1_1_1_4"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_1_2_1_1_1_4"/>
(  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_2_1_1_1_4_1"><b>suppose </b></a><a NAME="E1:14_1_1_2_1_1_1_4_1"/><i><font color="Green">E87</font></i>: 
 <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:14_1_1_2_1_1_1_4_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/><a NAME="E3:14_1_1_2_1_1_1_4_1"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:14_1_1_2_1_1_1_4_1"/><b>then </b>
<font color="Maroon">Co</font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font>,<span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span>,<span class="p2"><a href="pralg_2.html#K8">|.</a><span class="default"><font color="Maroon">OAF</font></span><a href="pralg_2.html#K8">.|</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_6.html#T85">FUNCT_6:85</a></i>;<br/><b>then consider </b><font color="Maroon">ss</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:14_1_1_2_1_1_1_4_1"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">ss</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Co</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ss</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ss</font> <a href="tarski.html#R1">c=</a>  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span>,<span class="p1"><a href="pralg_2.html#K8">|.</a><span class="default"><font color="Maroon">OAF</font></span><a href="pralg_2.html#K8">.|</a></span> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">y1</font> = <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/><a NAME="E8:14_1_1_2_1_1_1_4_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2_1_1_1_4_1_1"><b>proof </b></a><div class="add">

<i><font color="Green">E90</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#T10">PRALG_2:10</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E2:14_1_1_2_1_1_1_4_1_1"/>
<font color="Maroon">Co</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Co</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">ts</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:14_1_1_2_1_1_1_4_1_1"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">ts</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Co</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ts</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ts</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="pralg_2.html#K8">|.</a><span class="default"><font color="Maroon">OAF</font></span><a href="pralg_2.html#K8">.|</a></span> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E5:14_1_1_2_1_1_1_4_1_1"/>
for <font color="Olive">w</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> holds <br/><font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2_1_1_1_4_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:14_1_1_2_1_1_1_4_1_1_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:14_1_1_2_1_1_1_4_1_1_2"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">y</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:14_1_1_2_1_1_1_4_1_1_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="msalimit.html#E2:14_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:14_1_1_2_1_1_1_4_1_1_2"/>
<font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">MS</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font>
 
<b>by </b><i>, , , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E5:14_1_1_2_1_1_1_4_1_1_2"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">MS</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E6:14_1_1_2_1_1_1_4_1_1_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/>
<a NAME="E7:14_1_1_2_1_1_1_4_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:14_1_1_2_1_1_1_4_1_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="funct_6.html#K10">commute</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_6.html#T87">FUNCT_6:87</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as  <a href="funcop_1.html#V1">Function-yielding</a> <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E10:14_1_1_2_1_1_1_4_1_1_2"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">MS</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">yw</font> = <font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<b>reconsider </b><font color="Maroon">Pi</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E13:14_1_1_2_1_1_1_4_1_1_2"/>
<font color="Maroon">yw</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">ff</font> where <font color="Olive">ff</font> is    <a href="pralg_2.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<font color="Maroon">Pi</font></span>)</span> : for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>  st <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Olive">i</font>,<font color="Olive">j</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">Pi</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">ff</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">ff</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="msalimit.html#D1" target="_self">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon">jg</font> being    <a href="pralg_2.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<font color="Maroon">Pi</font></span>)</span><b> such that </b><br/><a NAME="E15:14_1_1_2_1_1_1_4_1_1_2"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">jg</font> <a href="hidden.html#R1">=</a> <font color="Maroon">yw</font> &amp; ( for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>  st <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Olive">i</font>,<font color="Olive">j</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">Pi</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">jg</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">jg</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> ) )
 ;<br/>
<a NAME="E16:14_1_1_2_1_1_1_4_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E17:14_1_1_2_1_1_1_4_1_1_2"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">yw</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<b>consider </b><font color="Maroon">U0</font> being    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E19:14_1_1_2_1_1_1_4_1_1_2"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">U0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font></span>)</span> )
 <b>by </b><i><a class="ref" href="pralg_2.html#D16">PRALG_2:def 16</a></i>;<br/>
<span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<b>hence </b><a NAME="E21:14_1_1_2_1_1_1_4_1_1_2"/>
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font>
 <b>by </b><i>, , <a class="txt" href="msalimit.html#E2:14_1_1"><i><font color="Green">E29</font></i></a>, , <a class="ref" href="funct_6.html#T86">FUNCT_6:86</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:14_1_1_2_1_1_1_4_1_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:14_1_1_2_1_1_1_4_1"/><i><font color="Green">E102</font></i>: 
for <font color="Olive">t</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span> holds <br/><font color="Maroon">Co</font> <a href="funct_1.html#K1">.</a> <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">t</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2_1_1_1_4_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:14_1_1_2_1_1_1_4_1_2"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font> <b>by </b><i><a class="ref" href="pralg_2.html#T18">PRALG_2:18</a></i>;<br/>
<a NAME="E3:14_1_1_2_1_1_1_4_1_2"/><i><font color="Green">E128</font></i>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">t</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#T18">PRALG_2:18</a></i>;<br/>
<a NAME="E4:14_1_1_2_1_1_1_4_1_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/>
<a NAME="E5:14_1_1_2_1_1_1_4_1_2"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:14_1_1_2_1_1_1_4_1_2"/><b>then </b>
<font color="Maroon">Co</font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font>,<span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span>,<span class="p2"><a href="pralg_2.html#K8">|.</a><span class="default"><font color="Maroon">OAF</font></span><a href="pralg_2.html#K8">.|</a></span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_6.html#T85">FUNCT_6:85</a></i>;<br/>
<b>then consider </b><font color="Maroon">ss</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:14_1_1_2_1_1_1_4_1_2"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">ss</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Co</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ss</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ss</font> <a href="tarski.html#R1">c=</a>  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span>,<span class="p1"><a href="pralg_2.html#K8">|.</a><span class="default"><font color="Maroon">OAF</font></span><a href="pralg_2.html#K8">.|</a></span> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">yt</font> = <font color="Maroon">Co</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="msalimit.html#E2:14_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_6.html#T84">FUNCT_6:84</a></i>;<br/>
<a NAME="E10:14_1_1_2_1_1_1_4_1_2"/>
<font color="Maroon">Co</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">t</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2_1_1_1_4_1_2_1"><b>proof </b></a><div class="add">

<i><font color="Green">E131</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">t</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#T10">PRALG_2:10</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E2:14_1_1_2_1_1_1_4_1_2_1"/>
<font color="Maroon">Co</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Co</font>
 
<b>by </b><i><a class="txt" href="msalimit.html#E2:14_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">ts</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:14_1_1_2_1_1_1_4_1_2_1"/><i><font color="Green">E132</font></i>: 
( <font color="Maroon">ts</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Co</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ts</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ts</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="pralg_2.html#K8">|.</a><span class="default"><font color="Maroon">OAF</font></span><a href="pralg_2.html#K8">.|</a></span> )
 <b>by </b><i><a class="txt" href="msalimit.html#E2:14_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E5:14_1_1_2_1_1_1_4_1_2_1"/><i><font color="Green">E133</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:14_1_1_2_1_1_1_4_1_2_1"/>
for <font color="Olive">w</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">t</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> holds <br/><font color="Maroon">yt</font> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">t</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2_1_1_1_4_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:14_1_1_2_1_1_1_4_1_2_1_2"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">t</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:14_1_1_2_1_1_1_4_1_2_1_2"/><b>then </b><i><font color="Green">E135</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:14_1_1_2_1_1_1_4_1_2_1_2"/>
<font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">MS</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font>
 
<b>by </b><i>, , , , <a class="ref" href="msalimit.html#D1" target="_self">Def1</a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E4:14_1_1_2_1_1_1_4_1_2_1_2"/><b>then </b><i><font color="Green">E136</font></i>: 
<font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">MS</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E5:14_1_1_2_1_1_1_4_1_2_1_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/>
<a NAME="E6:14_1_1_2_1_1_1_4_1_2_1_2"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:14_1_1_2_1_1_1_4_1_2_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="funct_6.html#K10">commute</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_6.html#T87">FUNCT_6:87</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as  <a href="funcop_1.html#V1">Function-yielding</a> <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E9:14_1_1_2_1_1_1_4_1_2_1_2"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">MS</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">yw</font> = <font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<b>reconsider </b><font color="Maroon">Pi</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E12:14_1_1_2_1_1_1_4_1_2_1_2"/>
<font color="Maroon">yw</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">ff</font> where <font color="Olive">ff</font> is    <a href="pralg_2.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<font color="Maroon">Pi</font></span>)</span> : for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>  st <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Olive">i</font>,<font color="Olive">j</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">Pi</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">ff</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">ff</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">jg</font> being    <a href="pralg_2.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<font color="Maroon">Pi</font></span>)</span><b> such that </b><br/><a NAME="E14:14_1_1_2_1_1_1_4_1_2_1_2"/><i><font color="Green">E138</font></i>: 
( <font color="Maroon">jg</font> <a href="hidden.html#R1">=</a> <font color="Maroon">yw</font> &amp; ( for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>  st <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Olive">i</font>,<font color="Olive">j</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">Pi</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">jg</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">jg</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> ) )
 ;<br/>
<a NAME="E15:14_1_1_2_1_1_1_4_1_2_1_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E16:14_1_1_2_1_1_1_4_1_2_1_2"/><b>then </b><i><font color="Green">E139</font></i>: 
<font color="Maroon">yw</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font>
 
<b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<b>consider </b><font color="Maroon">U0</font> being    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E18:14_1_1_2_1_1_1_4_1_2_1_2"/><i><font color="Green">E140</font></i>: 
( <font color="Maroon">U0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">t</font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font></span>)</span> )
 <b>by </b><i><a class="ref" href="pralg_2.html#D16">PRALG_2:def 16</a></i>;<br/>
<span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> = 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">t</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<b>hence </b><a NAME="E20:14_1_1_2_1_1_1_4_1_2_1_2"/>
<font color="Maroon">yt</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">t</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font>
 <b>by </b><i>, , <a class="ref" href="msalimit.html#D1" target="_self">Def1</a>, , <a class="ref" href="funct_6.html#T86">FUNCT_6:86</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:14_1_1_2_1_1_1_4_1_2_1"/>
<font color="Maroon">Co</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">t</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:14_1_1_2_1_1_1_4_1_2"/>
<font color="Maroon">Co</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font>
 <b>by </b><i>, <a class="ref" href="pralg_2.html#T10">PRALG_2:10</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:14_1_1_2_1_1_1_4_1"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">Co</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2_1_1_1_4_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:14_1_1_2_1_1_1_4_1_3"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/>
<a NAME="E2:14_1_1_2_1_1_1_4_1_3"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:14_1_1_2_1_1_1_4_1_3"/><b>then </b>
<font color="Maroon">Co</font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font>,<span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span>,<span class="p2"><a href="pralg_2.html#K8">|.</a><span class="default"><font color="Maroon">OAF</font></span><a href="pralg_2.html#K8">.|</a></span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_6.html#T85">FUNCT_6:85</a></i>;<br/>
<b>then consider </b><font color="Maroon">Co'</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:14_1_1_2_1_1_1_4_1_3"/><i><font color="Green">E143</font></i>: 
( <font color="Maroon">Co'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Co</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Co'</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Co'</font> <a href="tarski.html#R1">c=</a>  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span>,<span class="p1"><a href="pralg_2.html#K8">|.</a><span class="default"><font color="Maroon">OAF</font></span><a href="pralg_2.html#K8">.|</a></span> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E6:14_1_1_2_1_1_1_4_1_3"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">Co</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="msalimit.html#E2:14_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="pralg_2.html#T18">PRALG_2:18</a></i>;<br/>
<b>hence </b><a NAME="E7:14_1_1_2_1_1_1_4_1_3"/>
<font color="Maroon">Co</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">y1'</font> = <font color="Maroon">y1</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span> <b>by </b><i>, <a class="ref" href="pralg_2.html#T10">PRALG_2:10</a></i>;<br/><a NAME="E12:14_1_1_2_1_1_1_4_1"/><i><font color="Green">E145</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E13:14_1_1_2_1_1_1_4_1"/><i><font color="Green">E146</font></i>: 
 <a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font>,<font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, </i>;<br/><a NAME="E14:14_1_1_2_1_1_1_4_1"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">OPE</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K1">IFEQ</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>,<a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_2.html#K2">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3">Frege</a> <span class="p3">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pralg_2.html#D20">PRALG_2:def 20</a></i>;<br/><a NAME="E15:14_1_1_2_1_1_1_4_1"/><b>then </b><i><font color="Green">E148</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K2">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3">Frege</a> <span class="p3">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><i><font color="Green">E149</font></i>: <font color="Maroon">x1</font> = 
<font color="Maroon">OPE</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_2.html#K2">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3">Frege</a> <span class="p3">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, , <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_2.html#K3">Frege</a> <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="pralg_2.html#D6">PRALG_2:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span> <a href="pralg_1.html#K15">..</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="pralg_2.html#D8">PRALG_2:def 8</a></i>
;<br/><b>then </b><i><font color="Green">E150</font></i>: <font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="txt" href="msalimit.html#E2:14_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="pralg_1.html#D17">PRALG_1:def 17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font>
<b>by </b><i><a class="ref" href="pralg_2.html#T14">PRALG_2:14</a></i>
;<br/><a NAME="E18:14_1_1_2_1_1_1_4_1"/>
<span class="p1">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="msualg_3.html#K6">#</a> <font color="Maroon">y1'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2_1_1_1_4_1_6"><b>proof </b></a><div class="add">

<a NAME="E1:14_1_1_2_1_1_1_4_1_6"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/>
<a NAME="E2:14_1_1_2_1_1_1_4_1_6"/><b>then </b><i><font color="Green">E151</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:14_1_1_2_1_1_1_4_1_6"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="funct_6.html#K10">commute</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_6.html#T87">FUNCT_6:87</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as  <a href="funcop_1.html#V1">Function-yielding</a> <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E5:14_1_1_2_1_1_1_4_1_6"/>
<font color="Maroon">Co</font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font>,<span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span>,<span class="p2"><a href="pralg_2.html#K8">|.</a><span class="default"><font color="Maroon">OAF</font></span><a href="pralg_2.html#K8">.|</a></span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_6.html#T85">FUNCT_6:85</a></i>;<br/>
<b>then consider </b><font color="Maroon">ss</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:14_1_1_2_1_1_1_4_1_6"/><i><font color="Green">E152</font></i>: 
( <font color="Maroon">ss</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Co</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ss</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ss</font> <a href="tarski.html#R1">c=</a>  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span>,<span class="p1"><a href="pralg_2.html#K8">|.</a><span class="default"><font color="Maroon">OAF</font></span><a href="pralg_2.html#K8">.|</a></span> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y2</font> = <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E9:14_1_1_2_1_1_1_4_1_6"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">Co</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Co</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E10:14_1_1_2_1_1_1_4_1_6"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">Co</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Co</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">ts</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E12:14_1_1_2_1_1_1_4_1_6"/><i><font color="Green">E156</font></i>: 
( <font color="Maroon">ts</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Co</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ts</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ts</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="pralg_2.html#K8">|.</a><span class="default"><font color="Maroon">OAF</font></span><a href="pralg_2.html#K8">.|</a></span> )
 <b>by </b><i>, , <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Two</font> = <font color="Maroon">y2</font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i>, <a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<div><i><font color="Green">E158</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2_1_1_1_4_1_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:14_1_1_2_1_1_1_4_1_6_1"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">y2</font>
 ;<br/><a NAME="E2:14_1_1_2_1_1_1_4_1_6_1"/><b>then </b><i><font color="Green">E160</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>consider </b><font color="Maroon">ts'</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:14_1_1_2_1_1_1_4_1_6_1"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">ts'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Co</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ts'</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ts'</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="pralg_2.html#K8">|.</a><span class="default"><font color="Maroon">OAF</font></span><a href="pralg_2.html#K8">.|</a></span> )
 <b>by </b><i>, , <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><b>consider </b><font color="Maroon">sT</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:14_1_1_2_1_1_1_4_1_6_1"/><i><font color="Green">E164</font></i>: 
( <font color="Maroon">sT</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">sT</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">sT</font> <a href="tarski.html#R1">c=</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font>,<span class="p1"><a href="pralg_2.html#K8">|.</a><span class="default"><font color="Maroon">OAF</font></span><a href="pralg_2.html#K8">.|</a></span> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">yn</font> = <font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/><a NAME="E8:14_1_1_2_1_1_1_4_1_6_1"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">y1'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">yn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="txt" href="msalimit.html#E1:14_1_1_2_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_6.html#T86">FUNCT_6:86</a></i>;<br/><b>reconsider </b><font color="Maroon">Pi</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font> ;<br/><a NAME="E10:14_1_1_2_1_1_1_4_1_6_1"/><i><font color="Green">E167</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E11:14_1_1_2_1_1_1_4_1_6_1"/>
<font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">MS</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , <a class="txt" href="msalimit.html#E1:14_1_1_2_1"><i><font color="Green">E41</font></i></a>, , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E12:14_1_1_2_1_1_1_4_1_6_1"/><b>then </b>
<font color="Maroon">yn</font> <a href="hidden.html#R2">in</a> <font color="Maroon">MS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Pi</font>
 <b>by </b><i>, <a class="txt" href="msalimit.html#E8:14_1_1_2_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E13:14_1_1_2_1_1_1_4_1_6_1"/><b>then </b>
<font color="Maroon">yn</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is    <a href="pralg_2.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<font color="Maroon">Pi</font></span>)</span> : for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>  st <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Olive">i</font>,<font color="Olive">j</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">Pi</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> </span>}</span> 
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">yn'</font> being    <a href="pralg_2.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<font color="Maroon">Pi</font></span>)</span><b> such that </b><br/><a NAME="E15:14_1_1_2_1_1_1_4_1_6_1"/><i><font color="Green">E169</font></i>: 
( <font color="Maroon">yn'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">yn</font> &amp; ( for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>  st <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Olive">i</font>,<font color="Olive">j</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">Pi</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">yn'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">yn'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> ) )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2_1_1_1_4_1_6_1_1"><b>now </b></a><div class="add"><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="msualg_3.html#K6">#</a> <font color="Maroon">y1'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">yn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="txt" href="msalimit.html#E1:14_1_1_2_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="msalimit.html#E4:14_1_1_2_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="msalimit.html#E6:14_1_1_2_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="msualg_3.html#D8">MSUALG_3:def 8</a></i>
<br/>.= 
<font color="Maroon">yn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, <a class="txt" href="msalimit.html#E12:14_1_1_2_1"><i><font color="Green">E47</font></i></a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E17:14_1_1_2_1_1_1_4_1_6_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="msualg_3.html#K6">#</a> <font color="Maroon">y1'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="txt" href="msalimit.html#E1:14_1_1_2_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_6.html#T86">FUNCT_6:86</a></i>;<br/></div><b>end;</b></div>
<i><font color="Green">E170</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="msualg_3.html#K6">#</a> <font color="Maroon">y1'</font></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_3.html#T6">MSUALG_3:6</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">One</font> = <span class="p1">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="msualg_3.html#K6">#</a> <font color="Maroon">y1'</font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E17:14_1_1_2_1_1_1_4_1_6"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> holds <br/><font color="Maroon">One</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Two</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E18:14_1_1_2_1_1_1_4_1_6"/>
<span class="p1">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="msualg_3.html#K6">#</a> <font color="Maroon">y1'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="txt" href="msalimit.html#E1:14_1_1_2_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>


</div><b>end;</b></div><b>then </b><span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="msualg_3.html#K6">#</a> <font color="Maroon">y1'</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#T14">PRALG_2:14</a></i>
<br/>.= 
<font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="msalimit.html#E2:14_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="msalimit.html#D1" target="_self">Def1</a>, <a class="ref" href="pralg_1.html#D17">PRALG_1:def 17</a></i>
;<br/><b>hence </b><a NAME="E20:14_1_1_2_1_1_1_4_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , <a class="ref" href="msualg_3.html#D9">MSUALG_3:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_2_1_1_1_4_2"><b>suppose </b></a><a NAME="E1:14_1_1_2_1_1_1_4_2"/><i><font color="Green">E172</font></i>: 
 <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><i><font color="Green">E173</font></i>:  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">OAF</font></span>)</span> = 
the <a href="msualg_1.html#U5">Charact</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">OAF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_2.html#K14">OPS</a> <font color="Maroon">OAF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="pralg_2.html#T20">PRALG_2:20</a></i>
<br/>.= 
 <a href="cqc_lang.html#K1">IFEQ</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span>,<a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pralg_2.html#K2">Commute</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K3">Frege</a> <span class="p3">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#D20">PRALG_2:def 20</a></i>
<br/>.= 
 <a href="funct_6.html#K10">commute</a> <span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span>
<b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/><a NAME="E3:14_1_1_2_1_1_1_4_2"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">MS</font> <a href="funct_1.html#NK3" target="_self">*</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>reconsider </b><font color="Maroon">co</font> = <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/><a NAME="E5:14_1_1_2_1_1_1_4_2"/><i><font color="Green">E176</font></i>: 
<font color="Maroon">co</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K5">curry'</a> <span class="p2">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <span class="p3">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="funct_6.html#D12">FUNCT_6:def 12</a></i>;<br/><a NAME="E6:14_1_1_2_1_1_1_4_2"/><i><font color="Green">E177</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E7:14_1_1_2_1_1_1_4_2"/><i><font color="Green">E178</font></i>: 
for <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font> holds  <font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2_1_1_1_4_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>;<br/>

<b>reconsider </b><font color="Maroon">g</font> = <span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E2:14_1_1_2_1_1_1_4_2_2"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">d</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#T14">PRALG_2:14</a></i>;<br/>
<b>then </b> <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> = 
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">d</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, <a class="ref" href="pralg_2.html#T11">PRALG_2:11</a></i>
;<br/>
<a NAME="E4:14_1_1_2_1_1_1_4_2_2"/><b>then </b><i><font color="Green">E180</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i>, , <a class="ref" href="card_3.html#T19">CARD_3:19</a>, <a class="ref" href="relat_1.html#T62">RELAT_1:62</a></i>;<br/>
<a NAME="E5:14_1_1_2_1_1_1_4_2_2"/><b>then </b><i><font color="Green">E181</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_5.html#T45">FUNCT_5:45</a></i>;<br/>
<b>reconsider </b><font color="Maroon">co'</font> = <span class="p1">(<span class="default"><a href="funct_5.html#K5">curry'</a> <span class="p2">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <span class="p3">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i/>;<br/>
<i><font color="Green">E183</font></i>: <font color="Maroon">co</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> = 
<font color="Maroon">co'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font>
<b>by </b><i><a class="ref" href="funct_6.html#D12">FUNCT_6:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">d</font>,<font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="funct_5.html#T29">FUNCT_5:29</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, , <a class="ref" href="funct_5.html#T45">FUNCT_5:45</a></i>
;<br/>
<a NAME="E8:14_1_1_2_1_1_1_4_2_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K13">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>then </b><font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> = 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="pralg_2.html#T14">PRALG_2:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, , , <a class="ref" href="card_3.html#T19">CARD_3:19</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E10:14_1_1_2_1_1_1_4_2_2"/>
<font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E8:14_1_1_2_1_1_1_4_2"/><i><font color="Green">E184</font></i>: 
 <a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font>,<font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, </i>;<br/><b>set </b><font color="Maroon">F</font> =  <a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font>;<br/><a NAME="E9:14_1_1_2_1_1_1_4_2"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="msalimit.html#E2:14_1_1"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E10:14_1_1_2_1_1_1_4_2"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="msalimit.html#E2:14_1_1"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E11:14_1_1_2_1_1_1_4_2"/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="pralg_2.html#T11">PRALG_2:11</a></i>;<br/><b>then reconsider </b><font color="Maroon">E</font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E13:14_1_1_2_1_1_1_4_2"/><i><font color="Green">E188</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="msualg_3.html#K6">#</a> <font color="Maroon">E</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="msualg_3.html#D9">MSUALG_3:def 9</a></i>;<br/><a NAME="E14:14_1_1_2_1_1_1_4_2"/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="pralg_2.html#T11">PRALG_2:11</a></i>;<br/><b>hence </b><a NAME="E15:14_1_1_2_1_1_1_4_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="msalimit.html#D1" target="_self">Def1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E2:14_1_1_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is    <a href="pralg_2.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> : for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>  st <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Olive">i</font>,<font color="Olive">j</font></span>)</span> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:14_1_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E22:14_1_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>
 
;<br/>
<a NAME="E23:14_1_1_2_1"/><b>then </b><i><font color="Green">E189</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>;<br/>
<a NAME="E24:14_1_1_2_1"/>
 <a href="relset_1.html#K4">dom</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E25:14_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:14_1_1_2"/>
<font color="Maroon">C</font> <a href="msualg_2.html#R1">is_closed_on</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="msualg_2.html#D6">MSUALG_2:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:14_1_1"/><b>then </b><i><font color="Green">E190</font></i>: 
<font color="Maroon">C</font> is <a href="msualg_2.html#V3">opers_closed</a>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D7">MSUALG_2:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">L</font> =  <a href="pralg_2.html#K15">product</a> <font color="Maroon">OAF</font> as  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font> ;<br/>
<b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">C</font> as   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">L</font> ;<br/>
<b>set </b><font color="Maroon">MSA</font> = <font color="Maroon">L</font> <a href="msualg_2.html#K10">|</a> <font color="Maroon">C</font>;<br/>
<a NAME="E11:14_1_1"/><i><font color="Green">E193</font></i>: 
<font color="Maroon">L</font> <a href="msualg_2.html#K10">|</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G3">MSAlgebra</a>(# <font color="Maroon">C</font>,<span class="p1">(<span class="default"><a href="msualg_2.html#K4">Opers</a> <font color="Maroon">L</font>,<font color="Maroon">C</font></span>)</span> #)
 
<b>by </b><i>, <a class="ref" href="msualg_2.html#D16">MSUALG_2:def 16</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">OAF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>;<br/><a NAME="E1:14_1_1_3"/><i><font color="Green">E194</font></i>: 
<font color="Maroon">f</font> is    <a href="pralg_2.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>;<br/><b>thus </b><a NAME="E2:14_1_1_3"/>
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="msualg_2.html#K10">|</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> iff for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>  st <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Olive">i</font>,<font color="Olive">j</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_3_1"><b>proof </b></a><div class="add">

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:14_1_1_3_1_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="msualg_2.html#K10">|</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 ;<br/><a NAME="E2:14_1_1_3_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is    <a href="pralg_2.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<font color="Maroon">s</font></span>)</span> : for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>  st <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Olive">i</font>,<font color="Olive">j</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> </span>}</span> 
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="pralg_2.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<font color="Maroon">s</font></span>)</span><b> such that </b><br/><a NAME="E4:14_1_1_3_1_1"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E5:14_1_1_3_1_1"/><i><font color="Green">E196</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>  st <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Olive">i</font>,<font color="Olive">j</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>
 ;<br/><b>thus </b><a NAME="E6:14_1_1_3_1_1"/>
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>  st <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Olive">i</font>,<font color="Olive">j</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:14_1_1_3_1"/>
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>  st <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Olive">i</font>,<font color="Olive">j</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>
 ;<br/>

<a NAME="E3:14_1_1_3_1"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is    <a href="pralg_2.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">OAF</font>,<font color="Maroon">s</font></span>)</span> : for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>  st <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Olive">i</font>,<font color="Olive">j</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:14_1_1_3_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="msualg_2.html#K10">|</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E13:14_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being <a href="msualg_1.html#V4">strict</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="pralg_2.html#K15">product</a> <font color="Maroon">OAF</font> st <br/>for <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font><br/> for <font color="Olive">f</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">OAF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> holds <br/> ( <font color="Olive">f</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> iff for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>  st <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Olive">i</font>,<font color="Olive">j</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Olive">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> )
 ;<br/>


</div>
