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

<b>let </b><font color="Maroon">U1</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>, <font color="Maroon">U2</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/>

<b>assume </b><a NAME="E1:9"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">U1</font> is    <a href="unialg_2.html#M3">SubAlgebra</a> of <font color="Maroon">U2</font>
 ;<br/>

<b>let </b><font color="Maroon">B</font> be   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U2</font></span>)</span>;<br/>

<b>assume </b><a NAME="E2:9"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U1</font></span>)</span>
 ;<br/>

<a NAME="E3:9"/><i><font color="Green">E34</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U2</font></span>)</span> is   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E4:9"/><i><font color="Green">E49</font></i>: 
 <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">U1</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">U2</font>
 
<b>by </b><i>, </i>;<br/>
<b>let </b><font color="Maroon">o</font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">U2</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">o</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">U1</font></span>)</span> <b>by </b><i>, </i>;<br/>
<a NAME="E6:9"/><i><font color="Green">E50</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:9"/><i><font color="Green">E51</font></i>: 
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U2</font></span>)</span> <a href="hidden.html#R1">=</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="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U2</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 <span class="p2">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">U2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D9">MSUALG_1:def 9</a></i>;<br/>
<a NAME="E8:9"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> the <a href="msualg_1.html#U2">Arity</a> of <span class="p2">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">U1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</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="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U2</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 <span class="p2">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">U2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<a NAME="E1:9_1"/><i><font color="Green">E52</font></i>: 
 <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">U1</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">U2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:9_1"/>
<font color="Maroon">B</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_1.html#K10">MSAlg</a> <font color="Maroon">U2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<b>hence </b><a NAME="E3:9_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> the <a href="msualg_1.html#U2">Arity</a> of <span class="p2">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">U1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</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="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U2</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 <span class="p2">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">U2</font></span>)</span></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/>


</div><b>end;</b></div>
<a NAME="E9:9"/><b>then </b>
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U2</font></span>)</span></span>)</span>
 
<b>by </b><i>, , , , <a class="ref" href="msualg_1.html#D9">MSUALG_1:def 9</a></i>;<br/>
<a NAME="E10:9"/><b>then </b>
 <a href="relat_1.html#K1">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="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U2</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K3">Args</a> <font color="Maroon">a</font>,<span class="p3">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E11:9"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="relat_1.html#K1">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="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U2</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K3">Args</a> <font color="Maroon">a</font>,<span class="p3">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">a</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">X</font> =  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U1</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">Y</font> =  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">a</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U1</font></span>)</span></span>)</span>;<br/>
<a NAME="E12:9"/><i><font color="Green">E56</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">a</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p3">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U2</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3">Args</a> <font color="Maroon">a</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U1</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E13:9"/>
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="msualg_1.html#K5">Den</a> <font color="Maroon">a</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U1</font></span>)</span></span>)</span> holds <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"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">a</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><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="E1:9_2"/>
<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="msualg_1.html#K5">Den</a> <font color="Maroon">a</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U1</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:9_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:9_2"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K11">the_sort_of</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_1.html#T11">MSUALG_1:11</a></i>;<br/>
<a NAME="E4:9_2"/><i><font color="Green">E58</font></i>: 
(  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U1</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msualg_1.html#K8">MSSorts</a> <font color="Maroon">U1</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K9">MSCharact</a> <font color="Maroon">U1</font></span>)</span> #) &amp;  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U2</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msualg_1.html#K8">MSSorts</a> <font color="Maroon">U2</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K9">MSCharact</a> <font color="Maroon">U2</font></span>)</span> #) )
 
<b>by </b><i><a class="ref" href="msualg_1.html#D16">MSUALG_1:def 16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">cc</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">U2</font> <b>by </b><i>, <a class="ref" href="unialg_2.html#D8">UNIALG_2:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:9_2_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3">Opers</a> <font color="Maroon">U2</font>,<font color="Maroon">cc</font></span>)</span>
 ;<br/><a NAME="E2:9_2_1"/><i><font color="Green">E62</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3">Opers</a> <font color="Maroon">U2</font>,<font color="Maroon">cc</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon">cc</font> <a href="finseq_2.html#K3">*</a> </span>)</span>,<font color="Maroon">cc</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E3:9_2_1"/>
<span class="p1">(<span class="default"><a href="unialg_2.html#K3">Opers</a> <font color="Maroon">U2</font>,<font color="Maroon">cc</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3">Opers</a> <font color="Maroon">U2</font>,<font color="Maroon">cc</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:9_2_1"/>
<span class="p1">(<span class="default"><a href="unialg_2.html#K3">Opers</a> <font color="Maroon">U2</font>,<font color="Maroon">cc</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> is   <a href="funct_1.html#NM1">Function</a>
 <b>by </b><i>, <a class="ref" href="partfun1.html#T120">PARTFUN1:120</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> =  <a href="unialg_2.html#K3">Opers</a> <font color="Maroon">U2</font>,<font color="Maroon">cc</font> as  <a href="funcop_1.html#V1">Function-yielding</a> <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a></i>;<br/>
<b>set </b><font color="Maroon">ff1</font> = <span class="p1">(<span class="default"><a href="pboole.html#K8">*--&gt;</a> 0</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">U1</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">ff2</font> = <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#K6">signature</a> <font color="Maroon">U1</font></span>)</span></span>)</span> <a href="pboole.html#K9">--&gt;</a> 0;<br/>
<b>set </b><font color="Maroon">gg2</font> = <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#K6">signature</a> <font color="Maroon">U2</font></span>)</span></span>)</span> <a href="pboole.html#K9">--&gt;</a> 0;<br/>
<b>reconsider </b><font color="Maroon">gg1</font> = <span class="p1">(<span class="default"><a href="pboole.html#K8">*--&gt;</a> 0</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">U2</font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">U2</font></span>)</span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="msualg_1.html#T7">MSUALG_1:7</a></i>;<br/>
<a NAME="E9:9_2"/><i><font color="Green">E63</font></i>: 
 <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">U2</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G1">ManySortedSign</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<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#K6">signature</a> <font color="Maroon">U2</font></span>)</span></span>)</span>,<font color="Maroon">gg1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">U2</font></span>)</span></span>)</span> <a href="pboole.html#K9">--&gt;</a> 0</span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_1.html#T16">MSUALG_1:16</a></i>;<br/>
<i><font color="Green">E64</font></i>:  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U2</font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">U2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">U2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E11:9_2"/><b>then </b>
the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U2</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">op</font> = the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> as   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U2</font> ;<br/>
<a NAME="E13:9_2"/>
 <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U1</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msualg_1.html#K8">MSSorts</a> <font color="Maroon">U1</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K9">MSCharact</a> <font color="Maroon">U1</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_1.html#D16">MSUALG_1:def 16</a></i>;<br/>
<a NAME="E14:9_2"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="pboole.html#K9">--&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D14">MSUALG_1:def 14</a></i>;<br/>
<a NAME="E15:9_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T14">FUNCOP_1:14</a></i>;<br/>
<a NAME="E16:9_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> is   <a href="pboole.html#NM3">Component</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E17:9_2"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>
 
<b>by </b><i>, <a class="ref" href="msualg_1.html#D17">MSUALG_1:def 17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">ff1</font> = <span class="p1">(<span class="default"><a href="pboole.html#K8">*--&gt;</a> 0</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">U1</font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">U1</font></span>)</span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="msualg_1.html#T7">MSUALG_1:7</a></i>;<br/>
<a NAME="E19:9_2"/><i><font color="Green">E67</font></i>: 
 <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">U1</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G1">ManySortedSign</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<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#K6">signature</a> <font color="Maroon">U1</font></span>)</span></span>)</span>,<font color="Maroon">ff1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">U1</font></span>)</span></span>)</span> <a href="pboole.html#K9">--&gt;</a> 0</span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_1.html#T16">MSUALG_1:16</a></i>;<br/>
<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E21:9_2"/><i><font color="Green">E68</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">U2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E22:9_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E24:9_2"/>
<font color="Maroon">U1</font>,<font color="Maroon">U2</font> <a href="unialg_2.html#R1">are_similar</a> 
 
<b>by </b><i>, <a class="ref" href="unialg_2.html#T16">UNIALG_2:16</a></i>;<br/>
<a NAME="E25:9_2"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="msualg_1.html#K6">signature</a> <font color="Maroon">U1</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K6">signature</a> <font color="Maroon">U2</font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D2">UNIALG_2:def 2</a></i>;<br/>
<a NAME="E26:9_2"/><b>then </b><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">U1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">U1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E27:9_2"/><i><font color="Green">E71</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">U1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E28:9_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pboole.html#K8">*--&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E29:9_2"/><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"><span class="p2">(<span class="default"><a href="pboole.html#K8">*--&gt;</a> 0</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">U1</font></span>)</span></span>)</span>
 
<b>by </b><i>, , , , <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>
<a NAME="E30:9_2"/><b>then </b><i><font color="Green">E72</font></i>: 
the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">U1</font></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"><a href="pboole.html#K8">*--&gt;</a> 0</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#K6">signature</a> <font color="Maroon">U1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">sig</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">U1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, </i>;<br/>
<a NAME="E32:9_2"/><i><font color="Green">E74</font></i>: 
the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">U1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">sig</font> <a href="finseq_2.html#K2">|-&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="pboole.html#D20">PBOOLE:def 20</a></i>;<br/>
<a NAME="E33:9_2"/>
<font color="Maroon">U1</font>,<font color="Maroon">U2</font> <a href="unialg_2.html#R1">are_similar</a> 
 
<b>by </b><i>, <a class="ref" href="unialg_2.html#T16">UNIALG_2:16</a></i>;<br/>
<a NAME="E34:9_2"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="msualg_1.html#K6">signature</a> <font color="Maroon">U1</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K6">signature</a> <font color="Maroon">U2</font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D2">UNIALG_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">ar</font> = the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">U1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i/>;<br/>
<a NAME="E36:9_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">ar</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>
 
<b>by </b><i>, <a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a></i>;<br/>
<a NAME="E37:9_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sig</font> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/>
<a NAME="E38:9_2"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">op</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>
 
<b>by </b><i>, , <a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/>
<a NAME="E39:9_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E40:9_2"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3">Opers</a> <font color="Maroon">U2</font>,<font color="Maroon">cc</font></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D7">UNIALG_2:def 7</a></i>;<br/>
<a NAME="E41:9_2"/>
<font color="Maroon">cc</font> is <a href="unialg_2.html#V1">opers_closed</a>
 
<b>by </b><i>, <a class="ref" href="unialg_2.html#D8">UNIALG_2:def 8</a></i>;<br/>
<a NAME="E42:9_2"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">cc</font> <a href="unialg_2.html#R2">is_closed_on</a> <font color="Maroon">op</font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D5">UNIALG_2:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">U1</font></span>)</span> ;<br/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">a</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K9">MSCharact</a> <font color="Maroon">U1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D15">MSUALG_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="unialg_2.html#D8">UNIALG_2:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">op</font> <a href="unialg_2.html#K2">/.</a> <font color="Maroon">cc</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="unialg_2.html#D7">UNIALG_2:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">op</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">op</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">cc</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="mssublat.html#T1" target="_self">Th1</a>, <a class="ref" href="unialg_2.html#D6">UNIALG_2:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U5">Charact</a> of <span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="msualg_1.html#D15">MSUALG_1:def 15</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"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
;<br/>
<b>hence </b><a NAME="E45:9_2"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">a</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:9"/>
for <font color="Olive">a</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">U1</font></span>)</span>  st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> holds <br/> <a href="msualg_1.html#K5">Den</a> <font color="Olive">a</font>,<span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U2</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3">Args</a> <font color="Olive">a</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">U1</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>


</div>
