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

<b>let </b><font color="Maroon">SG</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">AG</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">SG</font>;<br/><b>let </b><font color="Maroon">C</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:34"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">AG</font> :  ex <font color="Olive">R</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font> st <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> </span>}</span> 
 ;<br/>

<b>let </b><font color="Maroon">F</font> be    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">C</font>,<font color="Maroon">SG</font>;<br/>

<b>assume </b><a NAME="E2:34"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> holds <br/><font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">c</font>
 ;<br/>

<b>set </b><font color="Maroon">I</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SG</font>;<br/>
<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font>;<br/><a NAME="E1:34_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">AG</font>
 <b>by </b><i><a class="ref" href="msualg_2.html#D20">MSUALG_2:def 20</a></i>;<br/><b>hence </b><a NAME="E2:34_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">CC</font> = <font color="Maroon">C</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>consider </b><font color="Maroon">T</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font>;<br/>
<b>reconsider </b><font color="Maroon">FF</font> = <font color="Maroon">F</font> as    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">CC</font>,<font color="Maroon">SG</font> ;<br/>
<b>set </b><font color="Maroon">pr</font> =  <a href="pralg_2.html#K15">product</a> <font color="Maroon">FF</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">t</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">SG</font> ex <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">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">t</font> ex <font color="Olive">A</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font> st <br/>( <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; ( for <font color="Olive">B</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font>  st <font color="Olive">A</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">B</font> holds <br/><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font> ) );<br/>
<b>consider </b><font color="Maroon">SOR</font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SG</font><b> such that </b><br/><a NAME="E8:34"/><i><font color="Green">E40</font></i>: 
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">SG</font> holds <br/>for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SOR</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> iff ( <font color="Olive">e</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">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">e</font>] ) )
 <b>from </b><i><a class="ref" href="pboole.html#S2">PBOOLE:sch 2</a>();<br/></i>
<a NAME="E9:34"/>
<font color="Maroon">SOR</font> is   <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">FF</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_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="pboole.html#D5">PBOOLE:def 5</a>,<a class="ref" href="pboole.html#D23">PBOOLE:def 23</a><br/>

<b>assume </b><a NAME="E1:34_2"/><i><font color="Green">E41</font></i>: 
<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">SG</font>
 ;<br/>

<b>let </b><font color="Maroon">e</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="E2:34_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SOR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E3:34_2"/><b>then </b>
<font color="Maroon">e</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">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:34_2"/>
<font color="Maroon">e</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">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">SOR</font> = <font color="Maroon">SOR</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">FF</font></span>)</span> ;<br/>
<a NAME="E11:34"/>
<font color="Maroon">SOR</font> is <a href="msualg_2.html#V3">opers_closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3"><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">SG</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_2.html#D7">MSUALG_2:def 7</a><br/><b>let </b><font color="Maroon">q</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>,<a class="ref" href="msualg_2.html#D6">MSUALG_2:def 6</a><br/>



<b>assume </b><a NAME="E1:34_3"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">q</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">FF</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">SOR</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">SG</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">r</font> =  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font>;<br/>
<b>set </b><font color="Maroon">ar</font> =  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font>;<br/>
<a NAME="E2:34_3"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">SOR</font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">FF</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E3:34_3"/>
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">FF</font></span>)</span> is   <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">FF</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E4:34_3"/><b>then </b><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">SOR</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">SG</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">FF</font></span>)</span> <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">SG</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 
<b>by </b><i>, <a class="ref" href="msualg_2.html#T3">MSUALG_2:3</a></i>;<br/>
<a NAME="E5:34_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">SOR</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">SG</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="tarski.html#R1">c=</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">FF</font></span>)</span>
 
;<br/>
<a NAME="E6:34_3"/><b>then </b><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">FF</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">SOR</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">SG</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">SOR</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">SG</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>, <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">FF</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T38">FUNCT_2:38</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q1</font> = <font color="Maroon">q</font> as    <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">FF</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="pralg_2.html#T10">PRALG_2:10</a></i>;<br/>
<b>consider </b><font color="Maroon">g</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:34_3"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</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">FF</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">SOR</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">SG</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span></span>)</span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</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">FF</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">SOR</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">SG</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E10:34_3"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">q</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"><a href="pralg_2.html#K15">product</a> <font color="Maroon">FF</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E11:34_3"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">SOR</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">SG</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</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"><a href="pralg_2.html#K15">product</a> <font color="Maroon">FF</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E13:34_3"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font>,<font color="Maroon">q</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3_1"><b>proof </b></a><div class="add">

<b>take </b>
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font>
;<br/>

<b>take </b>
<font color="Maroon">q1</font>
;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_3_1_1"/>
(  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_3_1_1_1"><b>suppose </b></a><a NAME="E1:34_3_1_1_1"/><i><font color="Green">E64</font></i>: 
 <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">A</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font>;<br/><b>take </b>
<font color="Maroon">A</font>
;<br/><b>thus </b><a NAME="E3:34_3_1_1_1"/>
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font>
 ;<br/><b>thus </b><a NAME="E4:34_3_1_1_1"/>
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/><b>let </b><font color="Maroon">B</font> be  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font>;<br/><b>assume </b><a NAME="E5:34_3_1_1_1"/>
<font color="Maroon">A</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">B</font>
 ;<br/><a NAME="E6:34_3_1_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">AG</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">AG</font> )
 <b>by </b><i><a class="ref" href="msualg_2.html#D20">MSUALG_2:def 20</a></i>;<br/><a NAME="E7:34_3_1_1_1"/><b>then </b>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font> )
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">iA</font> = <font color="Maroon">A</font>, <font color="Maroon">iB</font> = <font color="Maroon">B</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">CC</font> ;<br/><a NAME="E9:34_3_1_1_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">iA</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">iA</font> &amp; <font color="Maroon">iB</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">iB</font> )
 <b>by </b><i/>;<br/><a NAME="E10:34_3_1_1_1"/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="equation.html#T1" target="_self">Th1</a>, <a class="ref" href="pralg_2.html#T11">PRALG_2:11</a></i>;<br/><a NAME="E11:34_3_1_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E12:34_3_1_1_1"/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="equation.html#T1" target="_self">Th1</a>, <a class="ref" href="pralg_2.html#T11">PRALG_2:11</a></i>;<br/><a NAME="E13:34_3_1_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E14:34_3_1_1_1"/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">FF</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="equation.html#T1" target="_self">Th1</a>, <a class="ref" href="pralg_2.html#T11">PRALG_2:11</a></i>;<br/><a NAME="E15:34_3_1_1_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> = 
<span class="p1">(<span class="default"><a href="pralg_3.html#K2">const</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">FF</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iA</font>
<b>by </b><i>, <a class="ref" href="pralg_3.html#D1">PRALG_3:def 1</a></i>
<br/>.= 
 <a href="pralg_3.html#K2">const</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><font color="Maroon">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">iA</font></span>)</span>
<b>by </b><i><a class="ref" href="equation.html#T1" target="_self">Th1</a>, <a class="ref" href="pralg_3.html#T10">PRALG_3:10</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">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">iA</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="pralg_3.html#D1">PRALG_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">AG</font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, , </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">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">iB</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, , </i>
<br/>.= 
 <a href="pralg_3.html#K2">const</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><font color="Maroon">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">iB</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_3.html#D1">PRALG_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_3.html#K2">const</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">FF</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iB</font>
<b>by </b><i><a class="ref" href="equation.html#T1" target="_self">Th1</a>, <a class="ref" href="pralg_3.html#T10">PRALG_3:10</a></i>
<br/>.= 
<font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font>
<b>by </b><i>, , <a class="ref" href="pralg_3.html#D1">PRALG_3:def 1</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_3_1_1_2"><b>suppose </b></a><a NAME="E1:34_3_1_1_2"/><i><font color="Green">E115</font></i>: 
 <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">domar</font> =  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/><a NAME="E3:34_3_1_1_2"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">SOR</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">SG</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/><a NAME="E4:34_3_1_1_2"/><b>then </b>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">SOR</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>
 ;<br/><a NAME="E5:34_3_1_1_2"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">g</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">SOR</font> <a href="msualg_4.html#K6">*</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/><b>then </b><i><font color="Green">E117</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">SOR</font> <a href="msualg_4.html#K6">*</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="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
<b>by </b><i><a class="ref" href="card_3.html#T18">CARD_3:18</a></i>
<br/>.= 
<font color="Maroon">domar</font>
<b>by </b><i><a class="ref" href="msualg_3.html#T6">MSUALG_3:6</a></i>
;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">gn</font> being  <a href="funct_1.html#NM1">Function</a> ex <font color="Olive">A</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font> st <br/>( <font color="Olive">gn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">B</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font>  st <font color="Olive">A</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">B</font> holds <br/><font color="Olive">gn</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">gn</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font> ) &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> );<br/><a NAME="E7:34_3_1_1_2"/><i><font color="Green">E118</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">domar</font>  ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">AG</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3_1_1_2_2"><b>proof </b></a><div class="add">

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

<a NAME="E1:34_3_1_1_2_2"/><i><font color="Green">E130</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="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SG</font>
 
<b>by </b><i><a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/>
<a NAME="E2:34_3_1_1_2_2"/>
<font color="Maroon">g</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">SOR</font> <a href="msualg_4.html#K6">*</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="E3:34_3_1_1_2_2"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SOR</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#T23">FUNCT_1:23</a></i>;<br/>
<b>then consider </b><font color="Maroon">s</font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">SG</font>, <font color="Maroon">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">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>, <font color="Maroon">A</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font><b> such that </b><br/><a NAME="E5:34_3_1_1_2_2"/>
<font color="Maroon">s</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>and </b><br/><a NAME="E6:34_3_1_1_2_2"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E7:34_3_1_1_2_2"/><i><font color="Green">E132</font></i>: 
for <font color="Olive">B</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font>  st <font color="Maroon">A</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">B</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font>
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">A</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">AG</font> <b>by </b><i><a class="ref" href="msualg_2.html#D20">MSUALG_2:def 20</a></i>;<br/>
<b>take </b>
<font color="Maroon">b</font>
;<br/>

<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>take </b>
<font color="Maroon">A</font>
;<br/>

<b>thus </b><a NAME="E9:34_3_1_1_2_2"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E10:34_3_1_1_2_2"/>
for <font color="Olive">B</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font>  st <font color="Maroon">A</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">B</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font>
 <b>by </b><i><a class="txt" href="equation.html#E1:34"><i><font color="Green">E35</font></i></a></i>;<br/>

<b>thus </b><a NAME="E11:34_3_1_1_2_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">KK</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">domar</font>, <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">AG</font><b> such that </b><br/><a NAME="E9:34_3_1_1_2"/><i><font color="Green">E133</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">domar</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Maroon">KK</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i><a NAME="E10:34_3_1_1_2"/>
<font color="Maroon">KK</font> is    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">domar</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3_1_1_2_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:34_3_1_1_2_3"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">KK</font> <a href="hidden.html#R1">=</a> <font color="Maroon">domar</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">KK</font> = <font color="Maroon">KK</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">domar</font> ;<br/><a NAME="E12:34_3_1_1_2"/>
<font color="Maroon">KK</font> is    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">domar</font>,<font color="Maroon">SG</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3_1_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pralg_2.html#D12">PRALG_2:def 12</a><br/>

<b>assume </b><a NAME="E1:34_3_1_1_2_4"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">domar</font>
 ;<br/>

<b>then consider </b><font color="Maroon">gn</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:34_3_1_1_2_4"/><i><font color="Green">E134</font></i>: 
 ex <font color="Olive">A</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font> st <br/>( <font color="Maroon">gn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">B</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font>  st <font color="Olive">A</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">B</font> holds <br/><font color="Maroon">gn</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">gn</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font> ) &amp; <font color="Maroon">KK</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:34_3_1_1_2_4"/>
<font color="Maroon">KK</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> is  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">SG</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">KK</font> = <font color="Maroon">KK</font> as    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">domar</font>,<font color="Maroon">SG</font> ;<br/><a NAME="E14:34_3_1_1_2"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">domar</font>  ex <font color="Olive">C</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font> st <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">KK</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3_1_1_2_5"><b>proof </b></a><div class="add">

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

<a NAME="E1:34_3_1_1_2_5"/>
 ex <font color="Olive">gn</font> being  <a href="funct_1.html#NM1">Function</a> ex <font color="Olive">A</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font> st <br/>( <font color="Olive">gn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">B</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font>  st <font color="Olive">A</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">B</font> holds <br/><font color="Olive">gn</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">gn</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font> ) &amp; <font color="Maroon">KK</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:34_3_1_1_2_5"/>
 ex <font color="Olive">C</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font> st <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">KK</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">n</font>
 ;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">K</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font><b> such that </b><br/><a NAME="E16:34_3_1_1_2"/><i><font color="Green">E135</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">domar</font> holds  <font color="Maroon">KK</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">n</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">K</font>
 <b>by </b><i/>;<br/><b>take </b>
<font color="Maroon">K</font>
;<br/><b>thus </b><a NAME="E17:34_3_1_1_2"/>
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font>
 ;<br/><b>thus </b><a NAME="E18:34_3_1_1_2"/>
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/><b>let </b><font color="Maroon">B</font> be  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font>;<br/><b>assume </b><a NAME="E19:34_3_1_1_2"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">K</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">B</font>
 ;<br/><a NAME="E20:34_3_1_1_2"/>
( <font color="Maroon">K</font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">AG</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">AG</font> )
 <b>by </b><i><a class="ref" href="msualg_2.html#D20">MSUALG_2:def 20</a></i>;<br/><a NAME="E21:34_3_1_1_2"/><b>then </b>
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font> &amp; <font color="Maroon">K</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font> )
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">iB</font> = <font color="Maroon">B</font>, <font color="Maroon">iK</font> = <font color="Maroon">K</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">CC</font> ;<br/><a NAME="E23:34_3_1_1_2"/><i><font color="Green">E138</font></i>: 
( <font color="Maroon">iK</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">iK</font> &amp; <font color="Maroon">iB</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">iB</font> )
 <b>by </b><i/>;<br/><a NAME="E24:34_3_1_1_2"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">CC</font>,<span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p5">(<span class="default"><font color="Maroon">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">CC</font>, <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SG</font> : verum </span>}</span> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pralg_3.html#T15">PRALG_3:15</a></i>;<br/><a NAME="E25:34_3_1_1_2"/><b>then </b><i><font color="Green">E141</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iB</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">domar</font>
 <b>by </b><i/>;<br/><a NAME="E26:34_3_1_1_2"/><i><font color="Green">E142</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iK</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">domar</font>
 <b>by </b><i>, </i>;<br/><a NAME="E27:34_3_1_1_2"/><i><font color="Green">E143</font></i>: 
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iK</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iB</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iK</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3_1_1_2_6"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:34_3_1_1_2_6"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iK</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">gn</font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">A</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font><b> such that </b><br/><a NAME="E3:34_3_1_1_2_6"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">gn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E4:34_3_1_1_2_6"/><i><font color="Green">E146</font></i>: 
for <font color="Olive">B</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font>  st <font color="Maroon">A</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">B</font> holds <br/><font color="Maroon">gn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">gn</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font>
 <b>and </b><br/><a NAME="E5:34_3_1_1_2_6"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">KK</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="txt" href="equation.html#E3:34"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E6:34_3_1_1_2_6"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">KK</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">K</font>
 
<b>by </b><i>, <a class="txt" href="equation.html#E3:34"><i><font color="Green">E39</font></i></a>, </i>;<br/>
<a NAME="E7:34_3_1_1_2_6"/><b>then </b><i><font color="Green">E149</font></i>: 
<font color="Maroon">A</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">B</font>
 
<b>by </b><i><a class="txt" href="equation.html#E1:34"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="msualg_2.html#T7">MSUALG_2:7</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iB</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">gn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">iB</font>
<b>by </b><i><a class="txt" href="equation.html#E3:34"><i><font color="Green">E39</font></i></a>, , <a class="txt" href="equation.html#E1:34_3"><i><font color="Green">E42</font></i></a>, <a class="ref" href="pralg_3.html#T22">PRALG_3:22</a></i>
<br/>.= 
<font color="Maroon">gn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font>
<b>by </b><i><a class="txt" href="equation.html#E2:34_3"><i><font color="Green">E43</font></i></a>, </i>
<br/>.= 
<font color="Maroon">gn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">iK</font>
<b>by </b><i><a class="txt" href="equation.html#E2:34_3"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="equation.html#T3" target="_self">Th3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iK</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="equation.html#E3:34"><i><font color="Green">E39</font></i></a>, , <a class="txt" href="equation.html#E1:34_3"><i><font color="Green">E42</font></i></a>, <a class="ref" href="pralg_3.html#T22">PRALG_3:22</a></i>
;<br/>


</div><b>end;</b></div><a NAME="E28:34_3_1_1_2"/><i><font color="Green">E150</font></i>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iB</font> is    <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">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">iB</font></span>)</span>
 <b>by </b><i><a class="ref" href="equation.html#T1" target="_self">Th1</a>, <a class="ref" href="pralg_3.html#T21">PRALG_3:21</a></i>;<br/><a NAME="E29:34_3_1_1_2"/><i><font color="Green">E151</font></i>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iK</font> is    <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">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">iK</font></span>)</span>
 <b>by </b><i><a class="ref" href="equation.html#T1" target="_self">Th1</a>, <a class="ref" href="pralg_3.html#T21">PRALG_3:21</a></i>;<br/><b>thus </b><font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <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"><font color="Maroon">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">iK</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iK</font></span>)</span>
<b>by </b><i>, <a class="ref" href="equation.html#T1" target="_self">Th1</a>, <a class="ref" href="pralg_3.html#T23">PRALG_3:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">AG</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">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iK</font></span>)</span>
<b>by </b><i>, <a class="txt" href="equation.html#E1:34_3"><i><font color="Green">E42</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">AG</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">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iB</font></span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E2:34"><i><font color="Green">E38</font></i></a>, <a class="txt" href="equation.html#E3:34"><i><font color="Green">E39</font></i></a>, <a class="txt" href="equation.html#E8:34"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</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">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">iB</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iB</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">q1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font>
<b>by </b><i>, <a class="ref" href="equation.html#T1" target="_self">Th1</a>, <a class="ref" href="pralg_3.html#T23">PRALG_3:23</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E14:34_3"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SOR</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/>
<a NAME="E15:34_3"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SOR</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">SG</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E16:34_3"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">SOR</font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">SG</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:34"/><b>then </b><i><font color="Green">E152</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">FF</font></span>)</span> <a href="msualg_2.html#K10">|</a> <font color="Maroon">SOR</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G3">MSAlgebra</a>(# <font color="Maroon">SOR</font>,<span class="p1">(<span class="default"><a href="msualg_2.html#K4">Opers</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">FF</font></span>)</span>,<font color="Maroon">SOR</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_2.html#D16">MSUALG_2:def 16</a></i>;<br/>
<a NAME="E13:34"/>
<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">FF</font></span>)</span> <a href="msualg_2.html#K10">|</a> <font color="Maroon">SOR</font> is <a href="msualg_1.html#V5">non-empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4"><b>proof </b></a><div class="add">

<a NAME="E1:34_4"/>
<font color="Maroon">SOR</font> is <a href="relat_1.html#V2">non-empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4_1"><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="pboole.html#D16">PBOOLE:def 16</a><br/>

<b>assume </b><a NAME="E1:34_4_1"/><i><font color="Green">E153</font></i>: 
<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">SG</font>
 ;<br/>

<a NAME="E2:34_4_1"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:34_4_1"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">C</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> is <a href="msafree2.html#V3">finitely-generated</a> );<br/>
<b>consider </b><font color="Maroon">ST1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:34_4_1"/><i><font color="Green">E155</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ST1</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="equation.html#K2" target="_self">SuperAlgebraSet</a> <font color="Maroon">T</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">K</font> being   <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font> ex <font color="Olive">t</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">K</font> &amp; <font color="Olive">t</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Olive">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> );<br/>
<a NAME="E7:34_4_1"/><i><font color="Green">E156</font></i>: 
for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font> holds <br/> ex <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">c</font>,<font color="Olive">j</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:34_4_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font>
 ;<br/>

<b>then consider </b><font color="Maroon">C</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">AG</font><b> such that </b><br/><a NAME="E3:34_4_1_1"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E4:34_4_1_1"/><i><font color="Green">E158</font></i>: 
 ex <font color="Olive">R</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font> st <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">R</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font><b> such that </b><br/><a NAME="E6:34_4_1_1"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>
<a NAME="E7:34_4_1_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>
<b>then consider </b><font color="Maroon">t</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:34_4_1_1"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">t</font>
;<br/>

<b>take </b>
<font color="Maroon">R</font>
;<br/>

<b>take </b>
<font color="Maroon">t</font>
;<br/>

<b>thus </b><a NAME="E10:34_4_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="equation.html#T1" target="_self">Th1</a></i>;<br/>

<b>thus </b><a NAME="E11:34_4_1_1"/>
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">CC</font><b> such that </b><br/><a NAME="E9:34_4_1"/><i><font color="Green">E161</font></i>: 
for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">c</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">c</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>();<br/></i>
<b>set </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">ST1</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span>;<br/>
<a NAME="E10:34_4_1"/><i><font color="Green">E162</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">ST1</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">ST1</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E11:34_4_1"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">ST1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">CC</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:34_4_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ST1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">C</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font><b> such that </b><br/><a NAME="E3:34_4_1_2"/><i><font color="Green">E164</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp; <font color="Maroon">C</font> is <a href="msafree2.html#V3">finitely-generated</a> )
 <b>by </b><i/>;<br/>
<a NAME="E4:34_4_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">AG</font>
 
<b>by </b><i>, <a class="ref" href="msualg_2.html#D20">MSUALG_2:def 20</a></i>;<br/>
<b>hence </b><a NAME="E5:34_4_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>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">SG</font> <b>by </b><i/>;<br/>
<i><font color="Green">E185</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">ST1</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">ST1</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon">CC</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">ST1</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon">CC</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">ST1</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
<br/>.= 
<font color="Maroon">CC</font>
<b>by </b><i><a class="ref" href="equation.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
<br/>.= 
 <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">FF</font>,<font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<a NAME="E14:34_4_1"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">FF</font>,<font color="Maroon">s</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">ST1</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">a</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">FF</font>,<font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4_1_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:34_4_1_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">FF</font>,<font color="Maroon">s</font></span>)</span>
 ;<br/>

<a NAME="E2:34_4_1_4"/><b>then </b><i><font color="Green">E186</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">U0</font> being    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">SG</font><b> such that </b><br/><a NAME="E4:34_4_1_4"/><i><font color="Green">E187</font></i>: 
( <font color="Maroon">U0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">FF</font>,<font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 <b>by </b><i><a class="ref" href="pralg_2.html#D16">PRALG_2:def 16</a></i>;<br/>
<b>consider </b><font color="Maroon">K</font> being    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font>, <font color="Maroon">t</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:34_4_1_4"/><i><font color="Green">E188</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> )
 <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_4_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_4_1_4_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ST1</font> or  not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ST1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_4_1_4_1_1"><b>suppose </b></a><a NAME="E1:34_4_1_4_1_1"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ST1</font>
 ;<br/><div class="add"><a NAME="E2:34_4_1_4_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">ST1</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><a NAME="E3:34_4_1_4_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">ST1</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">ST1</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/><a NAME="E4:34_4_1_4_1_1"/><b>then </b><i><font color="Green">E190</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">ST1</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><a NAME="E5:34_4_1_4_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="equation.html#K2" target="_self">SuperAlgebraSet</a> <font color="Maroon">T</font>
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">M</font> being  <a href="msualg_1.html#V4">strict</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font><b> such that </b><br/><a NAME="E7:34_4_1_4_1_1"/><i><font color="Green">E191</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> &amp; <font color="Maroon">T</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">M</font> )
 <b>by </b><i/>;<br/><a NAME="E8:34_4_1_4_1_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">T</font> is    <a href="pboole.html#M4">ManySortedSubset</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">M</font>
 <b>by </b><i>, <a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/><a NAME="E9:34_4_1_4_1_1"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">T</font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/><a NAME="E10:34_4_1_4_1_1"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/><a NAME="E11:34_4_1_4_1_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 <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E12:34_4_1_4_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">ST1</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</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">FF</font>,<font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, , , <a class="txt" href="equation.html#E1:34"><i><font color="Green">E35</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_4_1_4_1_2"><b>suppose </b></a><a NAME="E1:34_4_1_4_1_2"/>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ST1</font>
 ;<br/><div class="add"><a NAME="E2:34_4_1_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">ST1</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>by </b><i>, , <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><b>hence </b><a NAME="E3:34_4_1_4_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">ST1</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</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">FF</font>,<font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E15:34_4_1"/><b>then </b><i><font color="Green">E192</font></i>: 
<font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">ST1</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">FF</font>,<font color="Maroon">s</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E16:34_4_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">ST1</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4_1_5"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">s</font>
;<br/>

<b>reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">ST1</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> as    <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">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <b>by </b><i>, <a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>;<br/>
<b>take </b>
<font color="Maroon">g1</font>
;<br/>

<b>take </b>
<font color="Maroon">T</font>
;<br/>

<b>thus </b><a NAME="E2:34_4_1_5"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 ;<br/>

<b>thus </b><a NAME="E3:34_4_1_5"/>
<font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">ST1</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">B</font> be  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font>;<br/>

<b>assume </b><a NAME="E4:34_4_1_5"/><i><font color="Green">E194</font></i>: 
<font color="Maroon">T</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">B</font>
 ;<br/>

<a NAME="E5:34_4_1_5"/>
<font color="Maroon">T</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#T6">MSUALG_2:6</a></i>;<br/>
<a NAME="E6:34_4_1_5"/><b>then </b>
<font color="Maroon">T</font> <a href="hidden.html#R2">in</a>  <a href="equation.html#K2" target="_self">SuperAlgebraSet</a> <font color="Maroon">T</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:34_4_1_5"/><b>then </b><i><font color="Green">E195</font></i>: 
<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ST1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:34_4_1_5"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="equation.html#K2" target="_self">SuperAlgebraSet</a> <font color="Maroon">T</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:34_4_1_5"/><b>then </b><i><font color="Green">E196</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ST1</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">T</font> = 
<span class="p1">(<span class="default"><font color="Maroon">ST1</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">T</font>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">ST1</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:34_4_1"/>
not <font color="Maroon">SOR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:34_4"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">FF</font></span>)</span> <a href="msualg_2.html#K10">|</a> <font color="Maroon">SOR</font></span>)</span> is <a href="relat_1.html#V2">non-empty</a>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_1.html#D8">MSUALG_1:def 8</a><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">PS</font> = <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">FF</font></span>)</span> <a href="msualg_2.html#K10">|</a> <font color="Maroon">SOR</font> as  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="pralg_2.html#K15">product</a> <font color="Maroon">F</font> ;<br/>
<b>take </b>
<font color="Maroon">PS</font>
;<br/>

<div><i><font color="Green">E197</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_5"><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">SG</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">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>;<br/><b>let </b><font color="Maroon">A</font> be  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font>;<br/><a NAME="E1:34_5"/><i><font color="Green">E198</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">AG</font>
 <b>by </b><i><a class="ref" href="msualg_2.html#D20">MSUALG_2:def 20</a></i>;<br/><a NAME="E2:34_5"/><i><font color="Green">E199</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">FF</font>,<font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">CC</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:34_5"/><b>then </b><i><font color="Green">E200</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">FF</font>,<font color="Maroon">s</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">U0</font> being    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">SG</font><b> such that </b><br/><a NAME="E5:34_5"/><i><font color="Green">E201</font></i>: 
<font color="Maroon">U0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E6:34_5"/><i><font color="Green">E202</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">FF</font>,<font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="pralg_2.html#D16">PRALG_2:def 16</a></i>;<br/><a NAME="E7:34_5"/>
<font color="Maroon">FF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E8:34_5"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> is    <a href="pboole.html#M4">ManySortedSubset</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">AG</font>
 <b>by </b><i>, <a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/><a NAME="E9:34_5"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">AG</font>
 <b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/><a NAME="E10:34_5"/><b>then </b><i><font color="Green">E203</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">AG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/><a NAME="E11:34_5"/>
<span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">FF</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">FF</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="E12:34_5"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</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">FF</font>,<font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><b>hence </b><a NAME="E13:34_5"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">AG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="equation.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">SG</font> ex <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">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> ex <font color="Olive">A</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font> st <br/>( <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>3</sub></font> &amp; <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; ( for <font color="Olive">B</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font>  st <font color="Olive">A</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">B</font> holds <br/><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font> ) &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font> );<br/>
<div><i><font color="Green">E204</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:34_6"/><i><font color="Green">E205</font></i>: 
<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">SG</font>
 ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:34_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">PS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/><b>then consider </b><font color="Maroon">s</font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">SG</font>, <font color="Maroon">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">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>, <font color="Maroon">A</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font><b> such that </b><br/><a NAME="E4:34_6"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E5:34_6"/><i><font color="Green">E207</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E6:34_6"/><i><font color="Green">E208</font></i>: 
for <font color="Olive">B</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font>  st <font color="Maroon">A</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">B</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font>
 <b>by </b><i>, , </i>;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font>;<br/><b>thus </b><a NAME="E7:34_6"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">AG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E8:34_6"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">x</font>,<font color="Maroon">i</font>]
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">BA</font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">PS</font>,<font color="Maroon">AG</font><b> such that </b><br/><a NAME="E18:34"/><i><font color="Green">E209</font></i>: 
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">SG</font> holds <br/> ex <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">PS</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">AG</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">BA</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">PS</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">x</font>,<font color="Olive">i</font>] ) )
 <b>from </b><i><a class="ref" href="mssubfam.html#S1">MSSUBFAM:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">BA</font>
;<br/>

<b>thus </b><a NAME="E19:34"/>
<font color="Maroon">BA</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">PS</font>,<font color="Maroon">AG</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D10">MSUALG_3:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_7"><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">SG</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D9">MSUALG_3:def 9</a><br/>

<b>assume </b><a NAME="E1:34_7"/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">PS</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

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

<b>set </b><font color="Maroon">r</font> =  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font>;<br/>
<b>set </b><font color="Maroon">ar</font> =  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">PS</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>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">AG</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> such that </b><br/><a NAME="E3:34_7"/><i><font color="Green">E211</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">BA</font> <a href="msualg_3.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>and </b><br/><a NAME="E4:34_7"/><i><font color="Green">E212</font></i>: 
for <font color="Olive">k</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">PS</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> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>,<font color="Olive">k</font>, <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font>]
 <b>by </b><i/>;<br/>
<a NAME="E5:34_7"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">PS</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">PS</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_9.html#T19">MSUALG_9:19</a></i>;<br/>
<b>then consider </b><font color="Maroon">s</font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">SG</font>, <font color="Maroon">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">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>, <font color="Maroon">A</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font><b> such that </b><br/><a NAME="E7:34_7"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font>
 <b>and </b><br/><a NAME="E8:34_7"/><i><font color="Green">E213</font></i>: 
<font color="Maroon">f</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>,<font color="Maroon">PS</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E9:34_7"/><i><font color="Green">E214</font></i>: 
for <font color="Olive">B</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font>  st <font color="Maroon">A</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">B</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font>
 <b>and </b><br/><a NAME="E10:34_7"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">g</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#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">PS</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_7_1"/>
(  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_7_1_1"><b>suppose </b></a><a NAME="E1:34_7_1_1"/><i><font color="Green">E216</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"><a NAME="E2:34_7_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">AG</font>
 <b>by </b><i><a class="ref" href="msualg_2.html#D20">MSUALG_2:def 20</a></i>;<br/><a NAME="E3:34_7_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">iA</font> = <font color="Maroon">A</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">CC</font> ;<br/><b>reconsider </b><font color="Maroon">ff1</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">FF</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i/>;<br/><a NAME="E6:34_7_1_1"/><i><font color="Green">E218</font></i>: 
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">PS</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="pralg_2.html#T11">PRALG_2:11</a></i>;<br/><a NAME="E7:34_7_1_1"/><b>then </b><i><font color="Green">E219</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E8:34_7_1_1"/>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="pralg_2.html#T11">PRALG_2:11</a></i>;<br/><a NAME="E9:34_7_1_1"/><b>then </b><i><font color="Green">E220</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E10:34_7_1_1"/><i><font color="Green">E221</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">FF</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="pralg_3.html#K2">const</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">FF</font></span>)</span>
 <b>by </b><i><a class="ref" href="pralg_3.html#D1">PRALG_3:def 1</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">BA</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">PS</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> = 
<font color="Maroon">ff1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_3.html#K2">const</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">FF</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iA</font>
<b>by </b><i>, <a class="txt" href="equation.html#E1:34"><i><font color="Green">E35</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
 <a href="pralg_3.html#K2">const</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><font color="Maroon">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">iA</font></span>)</span>
<b>by </b><i>, <a class="ref" href="pralg_3.html#T10">PRALG_3:10</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">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">iA</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="pralg_3.html#D1">PRALG_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">AG</font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">AG</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">BA</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="pralg_3.html#T12">PRALG_3:12</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_7_1_2"><b>suppose </b></a><a NAME="E1:34_7_1_2"/><i><font color="Green">E222</font></i>: 
 <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">domar</font> =  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">kn</font> being  <a href="funct_1.html#NM1">Function</a> ex <font color="Olive">A</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font> st <br/>( <font color="Olive">kn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">B</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font>  st <font color="Olive">A</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">B</font> holds <br/><font color="Olive">kn</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">kn</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font> ) &amp; <span class="p1">(<span class="default"><font color="Maroon">BA</font> <a href="funct_1.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="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">kn</font> <a href="hidden.html#R1">=</a> <font color="Olive">kn</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> );<br/><a NAME="E3:34_7_1_2"/><i><font color="Green">E224</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">domar</font>  ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">AG</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>,<font color="Olive">b</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_7_1_2_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:34_7_1_2_1"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SG</font>
 
<b>by </b><i><a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/>
<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">PS</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>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">AG</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> such that </b><br/><a NAME="E3:34_7_1_2_1"/><i><font color="Green">E225</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">BA</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> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">PS</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> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">x</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>] ) )
 <b>by </b><i/>;<br/>
<a NAME="E4:34_7_1_2_1"/>
<font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">PS</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">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_6.html#T2">MSUALG_6:2</a></i>;<br/>
<a NAME="E5:34_7_1_2_1"/><b>then </b>
<font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">PS</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="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">s</font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">SG</font>, <font color="Maroon">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">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>, <font color="Maroon">A</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font><b> such that </b><br/><a NAME="E7:34_7_1_2_1"/>
<font color="Maroon">s</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>and </b><br/><a NAME="E8:34_7_1_2_1"/><i><font color="Green">E226</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E9:34_7_1_2_1"/><i><font color="Green">E227</font></i>: 
( ( for <font color="Olive">B</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font>  st <font color="Maroon">A</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">B</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font> ) &amp; <font color="Maroon">g</font> <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="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">A</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">AG</font> <b>by </b><i><a class="ref" href="msualg_2.html#D20">MSUALG_2:def 20</a></i>;<br/>
<b>take </b>
<font color="Maroon">b</font>
;<br/>

<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>take </b>
<font color="Maroon">A</font>
;<br/>

<b>thus </b><a NAME="E11:34_7_1_2_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E12:34_7_1_2_1"/>
for <font color="Olive">B</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font>  st <font color="Maroon">A</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">B</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font>
 <b>by </b><i><a class="txt" href="equation.html#E1:34"><i><font color="Green">E35</font></i></a></i>;<br/>

<b>thus </b><a NAME="E13:34_7_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">BA</font> <a href="funct_1.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="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="txt" href="equation.html#E1:34"><i><font color="Green">E35</font></i></a></i>;<br/>

<b>thus </b><a NAME="E14:34_7_1_2_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">KK</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">domar</font>, <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">AG</font><b> such that </b><br/><a NAME="E5:34_7_1_2"/><i><font color="Green">E228</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">domar</font> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>,<font color="Maroon">KK</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i><a NAME="E6:34_7_1_2"/>
<font color="Maroon">KK</font> is    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">domar</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_7_1_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:34_7_1_2_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">KK</font> <a href="hidden.html#R1">=</a> <font color="Maroon">domar</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">KK</font> = <font color="Maroon">KK</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">domar</font> ;<br/><a NAME="E8:34_7_1_2"/>
<font color="Maroon">KK</font> is    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">domar</font>,<font color="Maroon">SG</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_7_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pralg_2.html#D12">PRALG_2:def 12</a><br/>

<b>assume </b><a NAME="E1:34_7_1_2_3"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">domar</font>
 ;<br/>

<a NAME="E2:34_7_1_2_3"/><b>then </b>
 ex <font color="Olive">kn</font> being  <a href="funct_1.html#NM1">Function</a> ex <font color="Olive">A</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font> st <br/>( <font color="Olive">kn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">B</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font>  st <font color="Olive">A</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">B</font> holds <br/><font color="Olive">kn</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">kn</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font> ) &amp; <span class="p1">(<span class="default"><font color="Maroon">BA</font> <a href="funct_1.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="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">kn</font> <a href="hidden.html#R1">=</a> <font color="Olive">kn</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font> &amp; <font color="Maroon">KK</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:34_7_1_2_3"/>
<font color="Maroon">KK</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> is  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">SG</font>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">KK</font> = <font color="Maroon">KK</font> as    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">domar</font>,<font color="Maroon">SG</font> ;<br/><a NAME="E10:34_7_1_2"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">domar</font>  ex <font color="Olive">C</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font> st <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">KK</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_7_1_2_4"><b>proof </b></a><div class="add">

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

<a NAME="E1:34_7_1_2_4"/>
 ex <font color="Olive">kn</font> being  <a href="funct_1.html#NM1">Function</a> ex <font color="Olive">A</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font> st <br/>( <font color="Olive">kn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">B</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font>  st <font color="Olive">A</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">B</font> holds <br/><font color="Olive">kn</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">kn</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font> ) &amp; <span class="p1">(<span class="default"><font color="Maroon">BA</font> <a href="funct_1.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="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">kn</font> <a href="hidden.html#R1">=</a> <font color="Olive">kn</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font> &amp; <font color="Maroon">KK</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:34_7_1_2_4"/>
 ex <font color="Olive">C</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font> st <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">KK</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">n</font>
 ;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">K</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font><b> such that </b><br/><a NAME="E12:34_7_1_2"/><i><font color="Green">E229</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">domar</font> holds  <font color="Maroon">KK</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">n</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">K</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">MAX</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font><b> such that </b><br/><a NAME="E14:34_7_1_2"/><i><font color="Green">E231</font></i>: 
<font color="Maroon">A</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">MAX</font>
 <b>and </b><br/><a NAME="E15:34_7_1_2"/><i><font color="Green">E232</font></i>: 
<font color="Maroon">K</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">MAX</font>
 <b>by </b><i/>;<br/><a NAME="E16:34_7_1_2"/><i><font color="Green">E233</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">FF</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E17:34_7_1_2"/>
<font color="Maroon">MAX</font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">AG</font>
 <b>by </b><i><a class="ref" href="msualg_2.html#D20">MSUALG_2:def 20</a></i>;<br/><a NAME="E18:34_7_1_2"/><b>then </b>
<font color="Maroon">MAX</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">iA</font> = <font color="Maroon">MAX</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">CC</font> ;<br/><b>reconsider </b><font color="Maroon">ff1</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">FF</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i/>;<br/><a NAME="E21:34_7_1_2"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iA</font> is    <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">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">iA</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="pralg_3.html#T21">PRALG_3:21</a></i>;<br/><a NAME="E22:34_7_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iA</font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><font color="Maroon">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">iA</font></span>)</span>
 ;<br/><a NAME="E23:34_7_1_2"/><b>then </b><i><font color="Green">E234</font></i>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iA</font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">MAX</font>
 <b>by </b><i/>;<br/><a NAME="E24:34_7_1_2"/><i><font color="Green">E235</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">CC</font>,<span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p5">(<span class="default"><font color="Maroon">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">CC</font>, <font color="Olive">m</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SG</font> : verum </span>}</span> </span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="pralg_3.html#T15">PRALG_3:15</a></i>;<br/><a NAME="E25:34_7_1_2"/><b>then </b><i><font color="Green">E236</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iA</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">domar</font>
 <b>by </b><i/>;<br/><a NAME="E26:34_7_1_2"/><i><font color="Green">E237</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">BA</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">domar</font>
 <b>by </b><i><a class="ref" href="msualg_6.html#T2">MSUALG_6:2</a></i>;<br/><a NAME="E27:34_7_1_2"/><i><font color="Green">E238</font></i>: 
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iA</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">BA</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iA</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_7_1_2_5"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:34_7_1_2_5"/><i><font color="Green">E239</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"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iA</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">arn</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">SG</font> <b>by </b><i><a class="txt" href="equation.html#E8:34"><i><font color="Green">E40</font></i></a>, <a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/>
<b>set </b><font color="Maroon">fs</font> = <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iA</font>;<br/>
<b>reconsider </b><font color="Maroon">fs</font> = <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iA</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">MAX</font> <b>by </b><i><a class="txt" href="equation.html#E2:34"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E4:34_7_1_2_5"/>
<font color="Maroon">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> <font color="Maroon">fs</font></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">n'</font> = <font color="Maroon">n</font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon">kn</font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">Ki</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font><b> such that </b><br/><a NAME="E7:34_7_1_2_5"/><i><font color="Green">E243</font></i>: 
<font color="Maroon">kn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E8:34_7_1_2_5"/><i><font color="Green">E244</font></i>: 
for <font color="Olive">B</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font>  st <font color="Maroon">Ki</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">B</font> holds <br/><font color="Maroon">kn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Ki</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kn</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font>
 <b>and </b><br/><a NAME="E9:34_7_1_2_5"/><i><font color="Green">E245</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">BA</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">arn</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">kn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Ki</font>
 <b>and </b><br/><a NAME="E10:34_7_1_2_5"/><i><font color="Green">E246</font></i>: 
<font color="Maroon">KK</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ki</font>
 <b>by </b><i>, <a class="txt" href="equation.html#E8:34"><i><font color="Green">E40</font></i></a>, </i>;<br/>
<a NAME="E11:34_7_1_2_5"/>
<font color="Maroon">Ki</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">K</font>
 
<b>by </b><i>, <a class="txt" href="equation.html#E8:34"><i><font color="Green">E40</font></i></a>, , </i>;<br/>
<a NAME="E12:34_7_1_2_5"/><b>then </b><i><font color="Green">E247</font></i>: 
<font color="Maroon">Ki</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">MAX</font>
 
<b>by </b><i>, <a class="ref" href="msualg_2.html#T7">MSUALG_2:7</a></i>;<br/>
<a NAME="E13:34_7_1_2_5"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">domar</font>
 
<b>by </b><i><a class="txt" href="equation.html#E3:34"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_2.html#T169">FUNCT_2:169</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">BA</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">BA</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n'</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E8:34"><i><font color="Green">E40</font></i></a>, , <a class="ref" href="msualg_3.html#D8">MSUALG_3:def 8</a></i>
<br/>.= 
<font color="Maroon">kn</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">KK</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E8:34"><i><font color="Green">E40</font></i></a>, , , , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">kn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">iA</font>
<b>by </b><i><a class="ref" href="equation.html#T3" target="_self">Th3</a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iA</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i>, <a class="txt" href="equation.html#E8:34"><i><font color="Green">E40</font></i></a>, , , <a class="ref" href="pralg_3.html#T22">PRALG_3:22</a></i>
;<br/><br/>


</div><b>end;</b></div><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">BA</font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">PS</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">MAX</font>
<b>by </b><i>, <a class="ref" href="equation.html#T1" target="_self">Th1</a>, , <a class="txt" href="equation.html#E1:34"><i><font color="Green">E35</font></i></a></i>
<br/>.= 
<font color="Maroon">ff1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">MAX</font>
<b>by </b><i>, </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">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">iA</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iA</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="pralg_3.html#T23">PRALG_3:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">MAX</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">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iA</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">AG</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">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">iA</font></span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E2:34"><i><font color="Green">E38</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">AG</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">BA</font> <a href="msualg_3.html#K6">#</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="txt" href="equation.html#E8:34"><i><font color="Green">E40</font></i></a>, , , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<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="msualg_3.html#D3">MSUALG_3:def 3</a><br/>

<b>assume </b><a NAME="E20:34"/>
<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">SG</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">SG</font> ;<br/>
<a NAME="E22:34"/><i><font color="Green">E248</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">FF</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">FF</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="E23:34"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">BA</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">AG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
;<br/>
<b>hence </b><a NAME="E24:34"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">BA</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">AG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

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

<b>assume </b><a NAME="E25:34"/><i><font color="Green">E249</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">AG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E26:34"/><i><font color="Green">E250</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">BA</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">PS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">ff</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E28:34"/><i><font color="Green">E252</font></i>: 
<font color="Maroon">ff</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">PS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E29:34"/>
<font color="Maroon">ff</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">FF</font>,<font color="Maroon">s</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="equation.html#T1" target="_self">Th1</a></i>;<br/>
<b>then consider </b><font color="Maroon">aa</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E31:34"/><i><font color="Green">E254</font></i>: 
( <font color="Maroon">ff</font> <a href="hidden.html#R1">=</a> <font color="Maroon">aa</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">aa</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">FF</font>,<font color="Maroon">s</font></span>)</span> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">FF</font>,<font color="Maroon">s</font></span>)</span> holds <br/><font color="Maroon">aa</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</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">FF</font>,<font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) )
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">L</font> being  <a href="relat_1.html#V2">V2</a> <a href="finset_1.html#V2">V19</a> <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">AG</font>;<br/>
<b>set </b><font color="Maroon">SY</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>;<br/>
<a NAME="E33:34"/><i><font color="Green">E256</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E34:34"/><i><font color="Green">E257</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<i><font color="Green">E258</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SG</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SG</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SG</font>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>
;<br/>
<a NAME="E36:34"/>
<font color="Maroon">L</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span> is    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SG</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_9"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:34_9"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SG</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">L</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span> as    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SG</font> ;<br/>
<i><font color="Green">E259</font></i>: <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>
<div><i><font color="Green">E260</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_11"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:34_11"/>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SG</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">s</font> )
 ;<br/><a NAME="E2:34_11"/><b>then </b>
not <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:34_11"/>
<font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/></div><b>end;</b></div>
<a NAME="E40:34"/>
<font color="Maroon">Y</font> is    <a href="pboole.html#M4">ManySortedSubset</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">AG</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a>,<a class="ref" href="pboole.html#D23">PBOOLE:def 23</a><br/>

<b>assume </b><a NAME="E1:34_12"/><i><font color="Green">E261</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SG</font>
 ;<br/>

<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="E2:34_12"/><i><font color="Green">E262</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_12_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_12_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">s</font> or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_12_1_1"><b>suppose </b></a><a NAME="E1:34_12_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><a NAME="E2:34_12_1_1"/><b>then </b><i><font color="Green">E263</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="equation.html#E1:34"><i><font color="Green">E35</font></i></a>, , </i>;<br/><a NAME="E3:34_12_1_1"/>
<font color="Maroon">L</font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">AG</font>
 <b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/><a NAME="E4:34_12_1_1"/><b>then </b>
<font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">AG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/><b>hence </b><a NAME="E5:34_12_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">AG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="equation.html#E2:34"><i><font color="Green">E38</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_12_1_2"><b>suppose </b></a><a NAME="E1:34_12_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:34_12_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">AG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , , <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>
<b>then reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">Y</font> as   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">AG</font> ;<br/>
<a NAME="E42:34"/>
<font color="Maroon">Y</font> is <a href="relat_1.html#V2">non-empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D16">PBOOLE:def 16</a><br/>

<b>assume </b><a NAME="E1:34_13"/><i><font color="Green">E264</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SG</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_13_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_13_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">s</font> or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_13_1_1"><b>suppose </b></a><a NAME="E1:34_13_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><a NAME="E2:34_13_1_1"/><b>then </b>
<font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="equation.html#E1:34"><i><font color="Green">E35</font></i></a>, </i>;<br/><b>hence </b><a NAME="E3:34_13_1_1"/>
not <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_13_1_2"><b>suppose </b></a><a NAME="E1:34_13_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:34_13_1_2"/>
not <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">Y</font> as  <a href="relat_1.html#V2">V2</a> <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">AG</font> ;<br/>
<a NAME="E44:34"/>
<font color="Maroon">Y</font> is <a href="finset_1.html#V2">locally-finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_14"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:34_14"/><i><font color="Green">E265</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SG</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_14_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_14_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">s</font> or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_14_1_1"><b>suppose </b></a><a NAME="E1:34_14_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><a NAME="E2:34_14_1_1"/><b>then </b>
<font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="equation.html#E1:34"><i><font color="Green">E35</font></i></a>, </i>;<br/><b>hence </b><a NAME="E3:34_14_1_1"/>
<font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, <a class="ref" href="pre_circ.html#D3">PRE_CIRC:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_14_1_2"><b>suppose </b></a><a NAME="E1:34_14_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:34_14_1_2"/>
<font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">Y</font> as  <a href="relat_1.html#V2">V2</a> <a href="finset_1.html#V2">V19</a> <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">AG</font> ;<br/>
<a NAME="E46:34"/>
<font color="Maroon">Y</font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D18">MSUALG_2:def 18</a></i>;<br/>
<a NAME="E47:34"/><b>then </b>
<font color="Maroon">Y</font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E48:34"/><b>then </b><i><font color="Green">E266</font></i>: 
<font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E49:34"/><i><font color="Green">E267</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E50:34"/><b>then </b><i><font color="Green">E268</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">Axx</font> being   <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font> st <br/>( <font color="Olive">Axx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Olive">Axx</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> );<br/>
<b>consider </b><font color="Maroon">WW</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E52:34"/><i><font color="Green">E271</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">WW</font> iff ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">a</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E53:34"/><i><font color="Green">E272</font></i>: 
<font color="Maroon">WW</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">CC</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_15"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:34_15"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">WW</font>
 ;<br/>

<b>hence </b><a NAME="E2:34_15"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font>
 <b>by </b><i><a class="txt" href="equation.html#E3:34"><i><font color="Green">E39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">XX</font> = <font color="Maroon">aa</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">WW</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">y</font></span>)</span>;<br/>
<a NAME="E54:34"/><i><font color="Green">E274</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">WW</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">WW</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<i><font color="Green">E275</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">aa</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">WW</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">aa</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">WW</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon">CC</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">WW</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon">CC</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">WW</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
<br/>.= 
<font color="Maroon">CC</font>
<b>by </b><i><a class="txt" href="equation.html#E8:34"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>
<a NAME="E56:34"/><i><font color="Green">E276</font></i>: 
for <font color="Olive">xx</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">CC</font>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">xx</font>] holds <br/><span class="p1">(<span class="default"><font color="Maroon">aa</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">WW</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">xx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_17"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:34_17"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">xx</font>]
 ;<br/>

<a NAME="E2:34_17"/><b>then </b><i><font color="Green">E277</font></i>: 
<font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">WW</font>
 
<b>by </b><i><a class="txt" href="equation.html#E3:34"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">aa</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">WW</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">xx</font> = 
<span class="p1">(<span class="default"><font color="Maroon">WW</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">xx</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E57:34"/><i><font color="Green">E278</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">FF</font>,<font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">CC</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E58:34"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">FF</font>,<font color="Maroon">s</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">aa</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">WW</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</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">FF</font>,<font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_18"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:34_18"/><i><font color="Green">E279</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">FF</font>,<font color="Maroon">s</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">C</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">AG</font><b> such that </b><br/><a NAME="E3:34_18"/><i><font color="Green">E280</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E4:34_18"/><i><font color="Green">E281</font></i>: 
 ex <font color="Olive">R</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font> st <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">R</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font><b> such that </b><br/><a NAME="E6:34_18"/><i><font color="Green">E282</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>
<a NAME="E7:34_18"/><i><font color="Green">E283</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">U0</font> being    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">SG</font><b> such that </b><br/><a NAME="E9:34_18"/><i><font color="Green">E284</font></i>: 
<font color="Maroon">U0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E10:34_18"/><i><font color="Green">E285</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">FF</font>,<font color="Maroon">s</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">s</font>
 <b>by </b><i><a class="ref" href="pralg_2.html#D16">PRALG_2:def 16</a></i>;<br/>
<a NAME="E11:34_18"/><i><font color="Green">E286</font></i>: 
<font color="Maroon">FF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font>
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_18_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_18_1"/>
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">x</font>] or  not <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">x</font>] )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_18_1_1"><b>suppose </b></a><a NAME="E1:34_18_1_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">x</font>]
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">Axx</font> being    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font><b> such that </b><br/><a NAME="E3:34_18_1_1"/><i><font color="Green">E287</font></i>: 
( <font color="Maroon">Axx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">Axx</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 ;<br/><b>thus </b><a NAME="E4:34_18_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">aa</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">WW</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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">FF</font>,<font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , , , <a class="ref" href="equation.html#T6" target="_self">Th6</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_18_1_2"><b>suppose </b></a><a NAME="E1:34_18_1_2"/>
 not <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">x</font>]
 ;<br/><div class="add"><a NAME="E2:34_18_1_2"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">WW</font>
 <b>by </b><i><a class="txt" href="equation.html#E3:34"><i><font color="Green">E39</font></i></a></i>;<br/><a NAME="E3:34_18_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">aa</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">WW</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">aa</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><b>hence </b><a NAME="E4:34_18_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">aa</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">WW</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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">FF</font>,<font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="equation.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">XX</font> = <font color="Maroon">aa</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">WW</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">y</font></span>)</span> as    <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">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</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">h</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">PS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">AG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font><b> such that </b><br/><a NAME="E61:34"/><i><font color="Green">E288</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">BA</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E62:34"/><i><font color="Green">E289</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">PS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">x</font>,<font color="Maroon">s</font>]
 <b>by </b><i/>;<br/>
<a NAME="E63:34"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">s</font>,<font color="Maroon">XX</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_19"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">s</font>
;<br/>

<b>take </b>
<font color="Maroon">XX</font>
;<br/>

<b>take </b><font color="Maroon">TT</font> =  <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">Y</font>;<br/>

<b>thus </b><a NAME="E1:34_19"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 ;<br/>

<b>thus </b><a NAME="E2:34_19"/>
<font color="Maroon">XX</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XX</font>
 ;<br/>

<b>let </b><font color="Maroon">B</font> be  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font>;<br/>

<b>assume </b><a NAME="E3:34_19"/><i><font color="Green">E291</font></i>: 
<font color="Maroon">TT</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">B</font>
 ;<br/>

<a NAME="E4:34_19"/><i><font color="Green">E292</font></i>: 
( <font color="Maroon">TT</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E5:34_19"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">TT</font> is    <a href="pboole.html#M4">ManySortedSubset</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E6:34_19"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">TT</font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E7:34_19"/><b>then </b><i><font color="Green">E293</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">TT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<b>thus </b><font color="Maroon">XX</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">TT</font> = 
<font color="Maroon">y</font>
<b>by </b><i>, , , </i>
<br/>.= 
<font color="Maroon">XX</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font>
<b>by </b><i><a class="txt" href="equation.html#E2:34"><i><font color="Green">E38</font></i></a>, , , </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E64:34"/><b>then </b><i><font color="Green">E294</font></i>: 
<font color="Maroon">XX</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SOR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i/>;<br/>
<a NAME="E65:34"/><b>then </b>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">XX</font>,<font color="Maroon">XX</font>,<font color="Maroon">s</font>]
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">t</font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">SG</font>, <font color="Maroon">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">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>, <font color="Maroon">A</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font><b> such that </b><br/><a NAME="E67:34"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>and </b><br/><a NAME="E68:34"/><i><font color="Green">E295</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XX</font>
 <b>and </b><br/><a NAME="E69:34"/><i><font color="Green">E296</font></i>: 
for <font color="Olive">B</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font>  st <font color="Maroon">A</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Olive">B</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font>
 <b>and </b><br/><a NAME="E70:34"/><i><font color="Green">E297</font></i>: 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">XX</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font>
 ;<br/>
<b>consider </b><font color="Maroon">MAX</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">AG</font><b> such that </b><br/><a NAME="E72:34"/><i><font color="Green">E298</font></i>: 
<font color="Maroon">A</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">MAX</font>
 <b>and </b><br/><a NAME="E73:34"/><i><font color="Green">E299</font></i>: 
 <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">Y</font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">MAX</font>
 <b>by </b><i/>;<br/>
<a NAME="E74:34"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">Y</font></span>)</span> is    <a href="pboole.html#M4">ManySortedSubset</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MAX</font>
 
<b>by </b><i>, <a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E75:34"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">Y</font></span>)</span> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MAX</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E76:34"/><b>then </b><i><font color="Green">E300</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">MAX</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E77:34"/><i><font color="Green">E301</font></i>: 
<font color="Maroon">MAX</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font>
 
<b>by </b><i/>;<br/>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">XX</font> = 
<font color="Maroon">XX</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">MAX</font>
<b>by </b><i>, , , <a class="ref" href="equation.html#T6" target="_self">Th6</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i><a class="txt" href="equation.html#E2:34"><i><font color="Green">E38</font></i></a>, , , </i>
;<br/>
<b>hence </b><a NAME="E79:34"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">BA</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="equation.html#T3" target="_self">Th3</a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div>
