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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>thus </b><a NAME="E1:22"/>
( <font color="Maroon">c<sub>2</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>2</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font> #) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:22_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font> #)
 <b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a><br/>

<b>thus </b><a NAME="E3:22_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_2.html#NM1">MSSubset</a> of <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font> #) st <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>1</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>2</sub></font> <a href="pboole.html#R6">=</a>  <a href="msualg_2.html#K4">Opers</a> <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font> #),<font color="Olive">b<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="msualg_2.html#NM1">MSSubset</a> of <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font> #);<br/>

<b>assume </b><a NAME="E1:22_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/>
<a NAME="E3:22_1_1"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>5</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>2</sub></font> <a href="pboole.html#R6">=</a>  <a href="msualg_2.html#K4">Opers</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="mssublat.html#E1:22_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="mssublat.html#E1:22_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E4:22_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="msualg_2.html#V3">opers_closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_2.html#D7">MSUALG_2:def 7</a><br/>

<i><font color="Green">E23</font></i>:  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> = 
the <a href="msualg_1.html#U5">Charact</a> of <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</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>6</sub></font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
<br/>.= 
 <a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>6</sub></font>,<a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font> #)
<b>by </b><i><a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
;<br/>
<a NAME="E2:22_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="msualg_2.html#R1">is_closed_on</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E3:22_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="msualg_2.html#D7">MSUALG_2:def 7</a></i>;<br/>
<a NAME="E3:22_1_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>6</sub></font>,<a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font> #)</span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">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>6</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E1:22_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="msualg_2.html#D6">MSUALG_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E4:22_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="msualg_2.html#R1">is_closed_on</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="msualg_2.html#D6">MSUALG_2:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:22_1_1"/>
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R6">=</a>  <a href="msualg_2.html#K4">Opers</a> <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font> #),<font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:22_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K4">Opers</a> <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font> #),<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:22_1_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<i><font color="Green">E23</font></i>:  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> = 
the <a href="msualg_1.html#U5">Charact</a> of <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</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>7</sub></font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
<br/>.= 
 <a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>7</sub></font>,<a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font> #)
<b>by </b><i><a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
;<br/>
<a NAME="E4:22_1_1_2_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="msualg_2.html#R1">is_closed_on</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E3:22_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="msualg_2.html#D7">MSUALG_2:def 7</a></i>;<br/>
<a NAME="E5:22_1_1_2_1"/><i><font color="Green">E25</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>7</sub></font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E4:22_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="msualg_2.html#D7">MSUALG_2:def 7</a></i>;<br/>
<span class="p1">(<span class="default"><a href="msualg_2.html#K4">Opers</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>7</sub></font> <a href="msualg_2.html#K3">/.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><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>7</sub></font>,<a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</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>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mssublat.html#E3:22_1_1_2_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="mssublat.html#E4:22_1_1_2_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="msualg_2.html#D8">MSUALG_2:def 8</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</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#E5:22_1_1_2_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="msualg_2.html#D8">MSUALG_2:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_2.html#K4">Opers</a> <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font> #),<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="msualg_2.html#D9">MSUALG_2:def 9</a></i>
;<br/>
<b>hence </b><a NAME="E7:22_1_1_2_1"/>
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>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K4">Opers</a> <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font> #),<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="mssublat.html#E1:22_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="mssublat.html#E1:22_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:22_1_1_2"/>
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R6">=</a>  <a href="msualg_2.html#K4">Opers</a> <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font> #),<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:22_1_1"/>
( <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>2</sub></font> <a href="pboole.html#R6">=</a>  <a href="msualg_2.html#K4">Opers</a> <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font> #),<font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="mssublat.html#E4:22_1_1"><i><font color="Green">E22</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E4:22_1"/>
verum
 ;<br/>


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

<b>assume </b><a NAME="E2:22"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font> #)
 ;<br/>

<b>hence </b><a NAME="E3:22"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a><br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>assume </b><a NAME="E4:22"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="msualg_2.html#NM1">MSSubset</a> of <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font> #) ;<br/>
<a NAME="E6:22"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>5</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>2</sub></font> <a href="pboole.html#R6">=</a>  <a href="msualg_2.html#K4">Opers</a> <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font> #),<font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="mssublat.html#E2:22"><i><font color="Green">E19</font></i></a>, <a class="txt" href="mssublat.html#E4:22"><i><font color="Green">E20</font></i></a>, <a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E7:22"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="msualg_2.html#V3">opers_closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_2.html#D7">MSUALG_2:def 7</a><br/>

<i><font color="Green">E23</font></i>:  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> = 
the <a href="msualg_1.html#U5">Charact</a> of <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</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>6</sub></font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
<br/>.= 
 <a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>6</sub></font>,<a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font> #)
<b>by </b><i><a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
;<br/>
<a NAME="E2:22_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="msualg_2.html#R1">is_closed_on</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E6:22"><i><font color="Green">E21</font></i></a>, <a class="ref" href="msualg_2.html#D7">MSUALG_2:def 7</a></i>;<br/>
<a NAME="E3:22_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">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>6</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E1:22_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="msualg_2.html#D6">MSUALG_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E4:22_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="msualg_2.html#R1">is_closed_on</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="msualg_2.html#D6">MSUALG_2:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:22"/>
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R6">=</a>  <a href="msualg_2.html#K4">Opers</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3"><b>proof </b></a><div class="add">

<a NAME="E1:22_3"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K4">Opers</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:22_3_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<i><font color="Green">E23</font></i>:  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font> = 
the <a href="msualg_1.html#U5">Charact</a> of <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</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>7</sub></font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
<br/>.= 
 <a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>7</sub></font>,<a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font> #)
<b>by </b><i><a class="ref" href="msualg_1.html#D11">MSUALG_1:def 11</a></i>
;<br/>
<a NAME="E4:22_3_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="msualg_2.html#R1">is_closed_on</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E6:22"><i><font color="Green">E21</font></i></a>, <a class="ref" href="msualg_2.html#D7">MSUALG_2:def 7</a></i>;<br/>
<a NAME="E5:22_3_1"/><i><font color="Green">E25</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>7</sub></font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E7:22"><i><font color="Green">E22</font></i></a>, <a class="ref" href="msualg_2.html#D7">MSUALG_2:def 7</a></i>;<br/>
<span class="p1">(<span class="default"><a href="msualg_2.html#K4">Opers</a> <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font> #),<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>7</sub></font> <a href="msualg_2.html#K3">/.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><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>7</sub></font>,<font color="Maroon">c<sub>3</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>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mssublat.html#E3:22_3_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="mssublat.html#E4:22_3_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="msualg_2.html#D8">MSUALG_2:def 8</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</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#E5:22_3_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="msualg_2.html#D8">MSUALG_2:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_2.html#K4">Opers</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="msualg_2.html#D9">MSUALG_2:def 9</a></i>
;<br/>
<b>hence </b><a NAME="E7:22_3_1"/>
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>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K4">Opers</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="mssublat.html#E2:22"><i><font color="Green">E19</font></i></a>, <a class="txt" href="mssublat.html#E4:22"><i><font color="Green">E20</font></i></a>, <a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:22_3"/>
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R6">=</a>  <a href="msualg_2.html#K4">Opers</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:22"/>
( <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>2</sub></font> <a href="pboole.html#R6">=</a>  <a href="msualg_2.html#K4">Opers</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="mssublat.html#E7:22"><i><font color="Green">E22</font></i></a></i>;<br/>


</div>
