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

<b>let </b><font color="Maroon" title="c1">U1</font>, <font color="Maroon" title="c2">U2</font> be   <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>;<br/>

<b>assume </b><a NAME="E1:8"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon" title="c1">U1</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c2">U2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">B</font> be   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c2">U2</font></span>)</span>;<br/>

<b>assume </b><a NAME="E2:8"/><i><font color="Green" title="E7">A2</font></i>: 
<font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">U1</font></span>)</span>
 ;<br/>

<a NAME="E3:8"/><i><font color="Green" title="E8">A3</font></i>: 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c2">U2</font></span>)</span> is   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c2">U2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E4:8"/><i><font color="Green" title="E9">A4</font></i>: 
 <a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c1">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c2">U2</font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E1:8"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="mssublat.html#T7" target="_self" title="MSSUBLAT:th.7">Th7</a></i>;<br/>
<b>let </b><font color="Maroon" title="c4">o</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c2">U2</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c4">o</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c1">U1</font></span>)</span> <b>by </b><i><a class="txt" href="mssublat.html#E1:8"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="mssublat.html#T7" target="_self" title="MSSUBLAT:th.7">Th7</a></i>;<br/>
<a NAME="E6:8"/><i><font color="Green" title="E10">A5</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c4">o</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c2">U2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c4">o</font>,<span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c2">U2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:8"/><i><font color="Green" title="E11">A6</font></i>: 
 <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c4">o</font>,<span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c2">U2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p3">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c2">U2</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <span class="p2">(<span class="default"><a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c2">U2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">o</font>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D9" title="MSUALG_1:def.9">MSUALG_1:def 9</a></i>;<br/>
<a NAME="E8:8"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <span class="p2">(<span class="default"><a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c1">U1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p3">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c2">U2</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <span class="p2">(<span class="default"><a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c2">U2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">o</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<a NAME="E1:8_1"/><i><font color="Green" title="E12">A7</font></i>: 
 <a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c1">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c2">U2</font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E1:8"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="mssublat.html#T7" target="_self" title="MSSUBLAT:th.7">Th7</a></i>;<br/>
<a NAME="E2:8_1"/>
<font color="Maroon" title="c3">B</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c2">U2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<b>hence </b><a NAME="E3:8_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <span class="p2">(<span class="default"><a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c1">U1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p3">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c2">U2</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <span class="p2">(<span class="default"><a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c2">U2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">o</font>
 <b>by </b><i><a class="txt" href="mssublat.html#E3:8"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="mssublat.html#E1:8_1"><i><font color="Green" title="E12">A7</font></i></a>, <a class="ref" href="msualg_2.html#T3" title="MSUALG_2:th.3">MSUALG_2:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:8"/><b>then </b>
 <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">U1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c4">o</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c2">U2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mssublat.html#E2:8"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="mssublat.html#E4:8"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="mssublat.html#E6:8"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="mssublat.html#E7:8"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="msualg_1.html#D9" title="MSUALG_1:def.9">MSUALG_1:def 9</a></i>;<br/>
<a NAME="E10:8"/><b>then </b>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c4">o</font>,<span class="p3">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c2">U2</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">U1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">U1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E11:8"/><b>then </b><i><font color="Green" title="E12">A8</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c4">o</font>,<span class="p3">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c2">U2</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">U1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">U1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">X</font> =  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">U1</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">Y</font> =  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">U1</font></span>)</span></span>)</span>;<br/>
<a NAME="E12:8"/><i><font color="Green" title="E13">A9</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">U1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c4">o</font>,<span class="p3">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c2">U2</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">U1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mssublat.html#E11:8"><i><font color="Green" title="E12">A8</font></i></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<a NAME="E13:8"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">U1</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c4">o</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c2">U2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">U1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:8_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">U1</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:8_2"/><b>then </b>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">U1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mssublat.html#E12:8"><i><font color="Green" title="E13">A9</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:8_2"/><b>then </b><i><font color="Green" title="E14">A10</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K11" title="MSUALG_1:func.11">the_sort_of</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">U1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_1.html#T11" title="MSUALG_1:th.11">MSUALG_1:11</a></i>;<br/>
<a NAME="E4:8_2"/><i><font color="Green" title="E15">A11</font></i>: 
(  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msualg_1.html#K8" title="MSUALG_1:func.8">MSSorts</a> <font color="Maroon" title="c1">U1</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K9" title="MSUALG_1:func.9">MSCharact</a> <font color="Maroon" title="c1">U1</font></span>)</span> #) &amp;  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c2">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msualg_1.html#K8" title="MSUALG_1:func.8">MSSorts</a> <font color="Maroon" title="c2">U2</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K9" title="MSUALG_1:func.9">MSCharact</a> <font color="Maroon" title="c2">U2</font></span>)</span> #) )
 
<b>by </b><i><a class="ref" href="msualg_1.html#D16" title="MSUALG_1:def.16">MSUALG_1:def 16</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">cc</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">U2</font> <b>by </b><i><a class="txt" href="mssublat.html#E1:8"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="unialg_2.html#D8" title="UNIALG_2:def.8">UNIALG_2:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">n</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:8_2_1"/><i><font color="Green" title="E16">A12</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3" title="UNIALG_2:func.3">Opers</a> <font color="Maroon" title="c2">U2</font>,<font color="Maroon" title="c9">cc</font></span>)</span>
 ;<br/><a NAME="E2:8_2_1"/><i><font color="Green" title="E17">A13</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3" title="UNIALG_2:func.3">Opers</a> <font color="Maroon" title="c2">U2</font>,<font color="Maroon" title="c9">cc</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">cc</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c9">cc</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E3:8_2_1"/>
<span class="p1">(<span class="default"><a href="unialg_2.html#K3" title="UNIALG_2:func.3">Opers</a> <font color="Maroon" title="c2">U2</font>,<font color="Maroon" title="c9">cc</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3" title="UNIALG_2:func.3">Opers</a> <font color="Maroon" title="c2">U2</font>,<font color="Maroon" title="c9">cc</font></span>)</span>
 <b>by </b><i><a class="txt" href="mssublat.html#E1:8_2_1"><i><font color="Green" title="E16">A12</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:8_2_1"/>
<span class="p1">(<span class="default"><a href="unialg_2.html#K3" title="UNIALG_2:func.3">Opers</a> <font color="Maroon" title="c2">U2</font>,<font color="Maroon" title="c9">cc</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 <b>by </b><i><a class="txt" href="mssublat.html#E2:8_2_1"><i><font color="Green" title="E17">A13</font></i></a>, <a class="ref" href="partfun1.html#T120" title="PARTFUN1:th.120">PARTFUN1:120</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c10">f</font> =  <a href="unialg_2.html#K3" title="UNIALG_2:func.3">Opers</a> <font color="Maroon" title="c2">U2</font>,<font color="Maroon" title="c9">cc</font> as  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="ref" href="funcop_1.html#D6" title="FUNCOP_1:def.6">FUNCOP_1:def 6</a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">ff1</font> = <span class="p1">(<span class="default"><a href="pboole.html#K8" title="PBOOLE:func.8">*--&gt;</a> 0</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c12">ff2</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span></span>)</span> <a href="pboole.html#K9" title="PBOOLE:func.9">--&gt;</a> 0;<br/>
<b>set </b><font color="Maroon" title="c13">gg2</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c2">U2</font></span>)</span></span>)</span> <a href="pboole.html#K9" title="PBOOLE:func.9">--&gt;</a> 0;<br/>
<b>reconsider </b><font color="Maroon" title="c14">gg1</font> = <span class="p1">(<span class="default"><a href="pboole.html#K8" title="PBOOLE:func.8">*--&gt;</a> 0</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c2">U2</font></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c2">U2</font></span>)</span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="ref" href="msualg_1.html#T7" title="MSUALG_1:th.7">MSUALG_1:7</a></i>;<br/>
<a NAME="E9:8_2"/><i><font color="Green" title="E16">A14</font></i>: 
 <a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c2">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#G1" title="MSUALG_1:aggr.1">ManySortedSign</a>(# <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c2">U2</font></span>)</span></span>)</span>,<font color="Maroon" title="c14">gg1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c2">U2</font></span>)</span></span>)</span> <a href="pboole.html#K9" title="PBOOLE:func.9">--&gt;</a> 0</span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_1.html#T16" title="MSUALG_1:th.16">MSUALG_1:16</a></i>;<br/>
<i><font color="Green" title="E17">A15</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c2">U2</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c2">U2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c2">U2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="unialg_1.html#D11" title="UNIALG_1:def.11">UNIALG_1:def 11</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c2">U2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">op</font> = the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c2">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> as   <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c2">U2</font> <b>by </b><i><a class="txt" href="mssublat.html#E9:8_2"><i><font color="Green" title="E16">A14</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E12:8_2"/>
 <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msualg_1.html#K8" title="MSUALG_1:func.8">MSSorts</a> <font color="Maroon" title="c1">U1</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K9" title="MSUALG_1:func.9">MSCharact</a> <font color="Maroon" title="c1">U1</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_1.html#D16" title="MSUALG_1:def.16">MSUALG_1:def 16</a></i>;<br/>
<a NAME="E13:8_2"/><b>then </b>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">U1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="pboole.html#K9" title="PBOOLE:func.9">--&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D14" title="MSUALG_1:def.14">MSUALG_1:def 14</a></i>;<br/>
<a NAME="E14:8_2"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">U1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T14" title="FUNCOP_1:th.14">FUNCOP_1:14</a></i>;<br/>
<a NAME="E15:8_2"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font> is   <a href="pboole.html#NM3" title="PBOOLE:NM.3">Component</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">U1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E16:8_2"/><b>then </b><i><font color="Green" title="E18">A16</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E3:8_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="ref" href="msualg_1.html#D17" title="MSUALG_1:def.17">MSUALG_1:def 17</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">ff1</font> = <span class="p1">(<span class="default"><a href="pboole.html#K8" title="PBOOLE:func.8">*--&gt;</a> 0</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="ref" href="msualg_1.html#T7" title="MSUALG_1:th.7">MSUALG_1:7</a></i>;<br/>
<a NAME="E18:8_2"/><i><font color="Green" title="E19">A17</font></i>: 
 <a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c1">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#G1" title="MSUALG_1:aggr.1">ManySortedSign</a>(# <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span></span>)</span>,<font color="Maroon" title="c16">ff1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span></span>)</span> <a href="pboole.html#K9" title="PBOOLE:func.9">--&gt;</a> 0</span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_1.html#T16" title="MSUALG_1:th.16">MSUALG_1:16</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E20:8_2"/><i><font color="Green" title="E20">A18</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c2">U2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c17">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E21:8_2"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c17">n</font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E9:8_2"><i><font color="Green" title="E16">A14</font></i></a>, <a class="txt" href="mssublat.html#E20:8_2"><i><font color="Green" title="E20">A18</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c18">a</font> = <font color="Maroon" title="c5">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E23:8_2"/>
<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font> <a href="unialg_2.html#R1" title="UNIALG_2:pred.1">are_similar</a> 
 
<b>by </b><i><a class="txt" href="mssublat.html#E1:8"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="unialg_2.html#T16" title="UNIALG_2:th.16">UNIALG_2:16</a></i>;<br/>
<a NAME="E24:8_2"/><b>then </b><i><font color="Green" title="E21">A19</font></i>: 
 <a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c1">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c2">U2</font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D2" title="UNIALG_2:def.2">UNIALG_2:def 2</a></i>;<br/>
<a NAME="E25:8_2"/><b>then </b><i><font color="Green" title="E22">A20</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mssublat.html#E9:8_2"><i><font color="Green" title="E16">A14</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E26:8_2"/><i><font color="Green" title="E23">A21</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E27:8_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pboole.html#K8" title="PBOOLE:func.8">*--&gt;</a> 0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E28:8_2"/><b>then </b>
<font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pboole.html#K8" title="PBOOLE:func.8">*--&gt;</a> 0</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mssublat.html#E9:8_2"><i><font color="Green" title="E16">A14</font></i></a>, <a class="txt" href="mssublat.html#E24:8_2"><i><font color="Green" title="E21">A19</font></i></a>, <a class="txt" href="mssublat.html#E25:8_2"><i><font color="Green" title="E22">A20</font></i></a>, <a class="txt" href="mssublat.html#E26:8_2"><i><font color="Green" title="E23">A21</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<a NAME="E29:8_2"/><b>then </b><i><font color="Green" title="E24">A22</font></i>: 
the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c1">U1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pboole.html#K8" title="PBOOLE:func.8">*--&gt;</a> 0</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mssublat.html#E18:8_2"><i><font color="Green" title="E19">A17</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">sig</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c1">U1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="mssublat.html#E25:8_2"><i><font color="Green" title="E22">A20</font></i></a>, <a class="txt" href="mssublat.html#E26:8_2"><i><font color="Green" title="E23">A21</font></i></a></i>;<br/>
<a NAME="E31:8_2"/><i><font color="Green" title="E25">A23</font></i>: 
the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c1">U1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">sig</font> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> 0
 
<b>by </b><i><a class="txt" href="mssublat.html#E29:8_2"><i><font color="Green" title="E24">A22</font></i></a>, <a class="ref" href="pboole.html#D20" title="PBOOLE:def.20">PBOOLE:def 20</a></i>;<br/>
<a NAME="E32:8_2"/>
<font color="Maroon" title="c1">U1</font>,<font color="Maroon" title="c2">U2</font> <a href="unialg_2.html#R1" title="UNIALG_2:pred.1">are_similar</a> 
 
<b>by </b><i><a class="txt" href="mssublat.html#E1:8"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="unialg_2.html#T16" title="UNIALG_2:th.16">UNIALG_2:16</a></i>;<br/>
<a NAME="E33:8_2"/><b>then </b><i><font color="Green" title="E26">A24</font></i>: 
 <a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c1">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <font color="Maroon" title="c2">U2</font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D2" title="UNIALG_2:def.2">UNIALG_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">ar</font> = the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c1">U1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="txt" href="mssublat.html#E31:8_2"><i><font color="Green" title="E25">A23</font></i></a></i>;<br/>
<a NAME="E35:8_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">ar</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E16:8_2"><i><font color="Green" title="E18">A16</font></i></a>, <a class="ref" href="msualg_1.html#D6" title="MSUALG_1:def.6">MSUALG_1:def 6</a></i>;<br/>
<a NAME="E36:8_2"/><b>then </b>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">sig</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E31:8_2"><i><font color="Green" title="E25">A23</font></i></a>, <a class="ref" href="finseq_2.html#T69" title="FINSEQ_2:th.69">FINSEQ_2:69</a></i>;<br/>
<a NAME="E37:8_2"/><b>then </b><i><font color="Green" title="E27">A25</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c15">op</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E9:8_2"><i><font color="Green" title="E16">A14</font></i></a>, <a class="txt" href="mssublat.html#E33:8_2"><i><font color="Green" title="E26">A24</font></i></a>, <a class="ref" href="unialg_1.html#D11" title="UNIALG_1:def.11">UNIALG_1:def 11</a></i>;<br/>
<a NAME="E38:8_2"/>
<font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c2">U2</font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E9:8_2"><i><font color="Green" title="E16">A14</font></i></a>, <a class="txt" href="mssublat.html#E10:8_2"><i><font color="Green" title="E17">A15</font></i></a></i>;<br/>
<a NAME="E39:8_2"/><b>then </b><i><font color="Green" title="E28">A26</font></i>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3" title="UNIALG_2:func.3">Opers</a> <font color="Maroon" title="c2">U2</font>,<font color="Maroon" title="c9">cc</font></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D7" title="UNIALG_2:def.7">UNIALG_2:def 7</a></i>;<br/>
<a NAME="E40:8_2"/>
<font color="Maroon" title="c9">cc</font> is <a href="unialg_2.html#V1" title="UNIALG_2:attr.1">opers_closed</a>
 
<b>by </b><i><a class="txt" href="mssublat.html#E1:8"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="unialg_2.html#D8" title="UNIALG_2:def.8">UNIALG_2:def 8</a></i>;<br/>
<a NAME="E41:8_2"/><b>then </b><i><font color="Green" title="E29">A27</font></i>: 
<font color="Maroon" title="c9">cc</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c15">op</font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D5" title="UNIALG_2:def.5">UNIALG_2:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">a</font> = <font color="Maroon" title="c18">a</font> as   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c1">U1</font></span>)</span> ;<br/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c21">a</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">U1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K9" title="MSUALG_1:func.9">MSCharact</a> <font color="Maroon" title="c1">U1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
<b>by </b><i><a class="txt" href="mssublat.html#E4:8_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="ref" href="msualg_1.html#D11" title="MSUALG_1:def.11">MSUALG_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D15" title="MSUALG_1:def.15">MSUALG_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
<b>by </b><i><a class="txt" href="mssublat.html#E1:8"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="unialg_2.html#D8" title="UNIALG_2:def.8">UNIALG_2:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">op</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c9">cc</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
<b>by </b><i><a class="txt" href="mssublat.html#E39:8_2"><i><font color="Green" title="E28">A26</font></i></a>, <a class="ref" href="unialg_2.html#D7" title="UNIALG_2:def.7">UNIALG_2:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">op</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c15">op</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c9">cc</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
<b>by </b><i><a class="txt" href="mssublat.html#E41:8_2"><i><font color="Green" title="E29">A27</font></i></a>, <a class="ref" href="unialg_2.html#D6" title="UNIALG_2:def.6">UNIALG_2:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c2">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
<b>by </b><i><a class="txt" href="mssublat.html#E37:8_2"><i><font color="Green" title="E27">A25</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c2">U2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
<b>by </b><i><a class="txt" href="mssublat.html#E4:8_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="ref" href="msualg_1.html#D15" title="MSUALG_1:def.15">MSUALG_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c4">o</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c2">U2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11" title="MSUALG_1:def.11">MSUALG_1:def 11</a></i>
;<br/>
<b>hence </b><a NAME="E44:8_2"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c4">o</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c2">U2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">U1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:8"/>
for <font color="Olive" title="b1">a</font> being  <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c1">U1</font></span>)</span>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o</font> holds <br/> <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b1">a</font>,<span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">U1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c4">o</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c2">U2</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Olive" title="b1">a</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">U1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mssublat.html#E12:8"><i><font color="Green" title="E13">A9</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>


</div>
