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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="realset2.html#V3">trivial</a> non <a href="msualg_1.html#V2">void</a> <a href="msualg_1.html#V6">segmental</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> as   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E2:20"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>4</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>3</sub></font> <a href="pboole.html#R6">=</a>  <a href="msualg_2.html#K4">Opers</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E3:20"/><i><font color="Green">E18</font></i>: 
 <a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1">(<span class="default"><a href="msualg_1.html#K11">the_sort_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K12">the_charact_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_1.html#D19">MSUALG_1:def 19</a></i>;<br/>
<a NAME="E4:20"/><i><font color="Green">E19</font></i>: 
 <a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1">(<span class="default"><a href="msualg_1.html#K11">the_sort_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K12">the_charact_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_1.html#D19">MSUALG_1:def 19</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>5</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E5:20"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> as   <a href="unialg_1.html#NM1">PFuncFinSequence</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="mssublat.html#E5:20"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E7:20"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E4:20"><i><font color="Green">E19</font></i></a>, <a class="ref" href="msualg_1.html#D18">MSUALG_1:def 18</a></i>;<br/>
<b>then </b><i><font color="Green">E22</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> = 
the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mssublat.html#E3:20"><i><font color="Green">E18</font></i></a>, <a class="ref" href="msualg_1.html#D18">MSUALG_1:def 18</a></i>
;<br/>
<a NAME="E9:20"/>
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 <span class="p1">(<span class="default"><a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> 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>9</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 <span class="p1">(<span class="default"><a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>9</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>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="unialg_2.html#NM5">operation</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:20_2"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="mssublat.html#E7:20"><i><font color="Green">E21</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:20_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="msualg_2.html#R1">is_closed_on</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E2:20"><i><font color="Green">E17</font></i></a>, <a class="ref" href="msualg_2.html#D7">MSUALG_2:def 7</a></i>;<br/>
<a NAME="E4:20_2"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="unialg_2.html#V1">opers_closed</a>
 
<b>by </b><i><a class="txt" href="mssublat.html#E5:20"><i><font color="Green">E20</font></i></a>, <a class="ref" href="mssublat.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E5:20_2"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="unialg_2.html#R2">is_closed_on</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D5">UNIALG_2:def 5</a></i>;<br/>
<a NAME="E6:20_2"/><i><font color="Green">E26</font></i>: 
 <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font> be    <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#K13">1-Alg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>assume </b><a NAME="E1:20_2_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><font color="Maroon">c<sub>11</sub></font> = 
the <a href="msualg_1.html#U5">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>
<b>by </b><i><a class="txt" href="mssublat.html#E3:20"><i><font color="Green">E18</font></i></a>, <a class="txt" href="mssublat.html#E1:20_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="msualg_1.html#D18">MSUALG_1:def 18</a></i>
<br/>.= 
 <a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
;<br/><b>then </b> <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> = 
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<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">c<sub>12</sub></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> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_1.html#T11">MSUALG_1:11</a></i>
;<br/><a NAME="E4:20_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K11">the_sort_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> </span>}</span> 
 <b>by </b><i><a class="txt" href="mssublat.html#E1:20_2_1_1"><i><font color="Green">E27</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K11">the_sort_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E6:20_2_1_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> )
 ;<br/><b>thus </b><a NAME="E7:20_2_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="mssublat.html#E6:20_2_1_1"><i><font color="Green">E28</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:20_2_1"/>
 <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="unialg_1.html#D10">UNIALG_1:def 10</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E27</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</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 <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> = 
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="msualg_1.html#D9">MSUALG_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="mssublat.html#E4:20"><i><font color="Green">E19</font></i></a>, <a class="txt" href="mssublat.html#E5:20"><i><font color="Green">E20</font></i></a>, <a class="txt" href="mssublat.html#E6:20_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="msualg_1.html#T11">MSUALG_1:11</a></i>
;<br/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> = 
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="mssublat.html#E4:20"><i><font color="Green">E19</font></i></a>, <a class="ref" href="msualg_1.html#D18">MSUALG_1:def 18</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="msualg_2.html#K3">/.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="mssublat.html#E2:20"><i><font color="Green">E17</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>12</sub></font>,<font color="Maroon">c<sub>2</sub></font></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>4</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 <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mssublat.html#E3:20_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="msualg_2.html#D8">MSUALG_2:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U5">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></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>4</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 <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
<br/>.= 
<font color="Maroon">c<sub>11</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>11</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mssublat.html#E3:20"><i><font color="Green">E18</font></i></a>, <a class="txt" href="mssublat.html#E1:20_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="mssublat.html#E7:20_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="msualg_1.html#D18">MSUALG_1:def 18</a></i>
;<br/>
<b>hence </b><a NAME="E9:20_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="unialg_2.html#K2">/.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="mssublat.html#E5:20_2"><i><font color="Green">E25</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="E10:20"/>
the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="unialg_2.html#K3">Opers</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="mssublat.html#E8:20"><i><font color="Green">E22</font></i></a>, <a class="ref" href="unialg_2.html#D7">UNIALG_2:def 7</a></i>;<br/>


</div>
