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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>assume </b><a NAME="E1:16"/><i><font color="Green">E13</font></i>: 
 <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font> as    <a href="msualg_1.html#L3">MSAlgebra</a> of  <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E3:16"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K9">MSCharact</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_1.html#D16">MSUALG_1:def 16</a></i>;<br/>
<a NAME="E4:16"/><i><font color="Green">E15</font></i>: 
 <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K9">MSCharact</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_1.html#D16">MSUALG_1:def 16</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></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">c<sub>2</sub></font></span>)</span></span>)</span> <a href="pboole.html#K9">--&gt;</a> 0;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></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">c<sub>2</sub></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">c<sub>2</sub></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="E6:16"/><i><font color="Green">E16</font></i>: 
 <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>8</sub></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">c<sub>2</sub></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>reconsider </b><font color="Maroon">c<sub>9</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>5</sub></font> as   <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">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="mssublat.html#E1:16"><i><font color="Green">E13</font></i></a>, <a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>10</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E8:16"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E9:16"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> is <a href="msualg_2.html#V3">opers_closed</a> &amp; the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R6">=</a>  <a href="msualg_2.html#K4">Opers</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="mssublat.html#E1:16"><i><font color="Green">E13</font></i></a>, <a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></font> as   <a href="unialg_1.html#NM1">PFuncFinSequence</a> of <font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="txt" href="mssublat.html#E8:16"><i><font color="Green">E17</font></i></a></i>;<br/>
<i><font color="Green">E19</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font> = 
 <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="mssublat.html#E3:16"><i><font color="Green">E14</font></i></a>, <a class="ref" href="msualg_1.html#D15">MSUALG_1:def 15</a></i>
<br/>.= 
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">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE: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">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mssublat.html#E6:16"><i><font color="Green">E16</font></i></a>, <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> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>2</sub></font></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> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E12:16"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a> <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="unialg_2.html#K2">/.</a> <font color="Maroon">c<sub>10</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><a NAME="E1:16_2"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/>

<a NAME="E2:16_2"/>
<font color="Maroon">c<sub>10</sub></font> is <a href="unialg_2.html#V1">opers_closed</a>
 
<b>by </b><i><a class="txt" href="mssublat.html#E1:16"><i><font color="Green">E13</font></i></a>, <a class="txt" href="mssublat.html#E8:16"><i><font color="Green">E17</font></i></a>, <a class="ref" href="mssublat.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E3:16_2"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="unialg_2.html#R2">is_closed_on</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D5">UNIALG_2:def 5</a></i>;<br/>
<a NAME="E4:16_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U5">Charact</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="mssublat.html#E4:16"><i><font color="Green">E15</font></i></a>, <a class="txt" href="mssublat.html#E11:16"><i><font color="Green">E19</font></i></a>, <a class="txt" href="mssublat.html#E1:16_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="msualg_1.html#D15">MSUALG_1:def 15</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></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">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E6:16_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="msualg_2.html#R1">is_closed_on</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E9:16"><i><font color="Green">E18</font></i></a>, <a class="ref" href="msualg_2.html#D7">MSUALG_2:def 7</a></i>;<br/>
<a NAME="E7:16_2"/><i><font color="Green">E23</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <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">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:16_2"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <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">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pboole.html#K8">*--&gt;</a> 0</span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:16_2_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <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">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="mssublat.html#E11:16"><i><font color="Green">E19</font></i></a>, <a class="txt" href="mssublat.html#E1:16_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="mssublat.html#E7:16_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/>

<a NAME="E2:16_2_1"/><i><font color="Green">E26</font></i>: 
 <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="E3:16_2_1"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E1:16_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="mssublat.html#E1:16_2_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/>
<b>hence </b><a NAME="E4:16_2_1"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pboole.html#K8">*--&gt;</a> 0</span>)</span>
 <b>by </b><i><a class="txt" href="mssublat.html#E2:16_2_1"><i><font color="Green">E26</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:16_2"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>14</sub></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">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>
<b>then </b><i><font color="Green">E26</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <span class="p2">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pboole.html#K8">*--&gt;</a> 0</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mssublat.html#E6:16"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</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="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mssublat.html#E9:16_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</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="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mssublat.html#E1:16_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="mssublat.html#E8:16_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D20">PBOOLE:def 20</a></i>
;<br/>
<a NAME="E11:16_2"/>
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E12:16_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> 0 is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="mssublat.html#E6:16"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finseq_2.html#T77">FINSEQ_2:77</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="finseq_2.html#K2">|-&gt;</a> 0 as    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E14:16_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <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">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E3:16"><i><font color="Green">E14</font></i></a>, <a class="ref" href="msualg_1.html#D14">MSUALG_1:def 14</a></i>;<br/>
<a NAME="E15:16_2"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font> <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>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E16:16_2"/><b>then </b><i><font color="Green">E28</font></i>: 
0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E17:16_2"/><i><font color="Green">E29</font></i>: 
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<i><font color="Green">E30</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <span class="p2">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> = 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mssublat.html#E10:16_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="pboole.html#D19">PBOOLE:def 19</a></i>
<br/>.= 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p4">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="pboole.html#K9">--&gt;</a> 0</span>)</span></span>)</span>

<br/>.= 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="pboole.html#K9">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mssublat.html#E16:16_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funcop_1.html#T23">FUNCOP_1:23</a></i>
<br/>.= 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mssublat.html#E8:16"><i><font color="Green">E17</font></i></a>, <a class="txt" href="mssublat.html#E14:16_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="mssublat.html#E17:16_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
 <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="card_3.html#T20">CARD_3:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="finseq_2.html#T111">FINSEQ_2:111</a></i>
;<br/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> = 
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="mssublat.html#E3:16"><i><font color="Green">E14</font></i></a>, <a class="ref" href="msualg_1.html#D15">MSUALG_1:def 15</a></i>
<br/>.= 
<font color="Maroon">c<sub>14</sub></font> <a href="msualg_2.html#K3">/.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="mssublat.html#E9:16"><i><font color="Green">E18</font></i></a>, <a class="ref" href="msualg_2.html#D9">MSUALG_2:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <span class="p3">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mssublat.html#E6:16_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="msualg_2.html#D8">MSUALG_2:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K9">MSCharact</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <span class="p3">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mssublat.html#E4:16"><i><font color="Green">E15</font></i></a>, <a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mssublat.html#E1:16_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="mssublat.html#E18:16_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="msualg_1.html#D15">MSUALG_1:def 15</a></i>
;<br/>
<b>hence </b><a NAME="E20:16_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="unialg_2.html#K2">/.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="mssublat.html#E3:16_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="unialg_2.html#D6">UNIALG_2:def 6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:16"/>
the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_2.html#K3">Opers</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="mssublat.html#E11:16"><i><font color="Green">E19</font></i></a>, <a class="ref" href="unialg_2.html#D7">UNIALG_2:def 7</a></i>;<br/>


</div>
