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<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> 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>reconsider </b><font color="Maroon">c<sub>3</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>2</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>2</sub></font> #) as  <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>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>2</sub></font> as  <a href="relat_1.html#V2">V3</a> <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="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</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="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E4:20"/><i><font color="Green">E8</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is   <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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E1:20_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/><a NAME="E3:20_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="msualg_2.html#V3">opers_closed</a>
 <b>by </b><i><a class="txt" href="msafree2.html#E1:20_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/><div><i><font color="Green">E11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:20_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="msualg_2.html#R1">is_closed_on</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="msafree2.html#E3:20_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="msualg_2.html#D7">MSUALG_2:def 7</a></i>;<br/><a NAME="E2:20_1_1"/><b>then </b><i><font color="Green">E12</font></i>: 
 <a href="relset_1.html#K5">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>9</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>2</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>2</sub></font> #)</span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>8</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>9</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</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>9</sub></font>
 <b>by </b><i><a class="ref" href="msualg_2.html#D6">MSUALG_2:def 6</a></i>;<br/> <a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</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>2</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>

<br/>.= 
 <a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>9</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>2</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>2</sub></font> #)

;<br/><b>hence </b><a NAME="E4:20_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="msualg_2.html#R1">is_closed_on</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="msafree2.html#E2:20_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="msualg_2.html#D6">MSUALG_2:def 6</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:20_1"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="msualg_2.html#V3">opers_closed</a>
 <b>by </b><i><a class="ref" href="msualg_2.html#D7">MSUALG_2:def 7</a></i>;<br/><a NAME="E6:20_1"/><i><font color="Green">E12</font></i>: 
the <a href="msualg_1.html#U5">Charact</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <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>8</sub></font>
 <b>by </b><i><a class="txt" href="msafree2.html#E1:20_1"><i><font color="Green">E9</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>9</sub></font> =  <a href="msualg_2.html#K4">Opers</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> as    <a href="pboole.html#M3">ManySortedFunction</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</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>,<font color="Maroon">c<sub>8</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> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:20_1_2"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="msualg_2.html#R1">is_closed_on</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="msafree2.html#E4:20_1"><i><font color="Green">E11</font></i></a></i>;<br/><a NAME="E2:20_1_2"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="msualg_2.html#R1">is_closed_on</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="msafree2.html#E3:20_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="msualg_2.html#D7">MSUALG_2:def 7</a></i>;<br/><i><font color="Green">E15</font></i>:  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</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>2</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>

<br/>.= 
 <a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>10</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>2</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>2</sub></font> #)

;<br/><b>thus </b><font color="Maroon">c<sub>9</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> = 
<font color="Maroon">c<sub>10</sub></font> <a href="msualg_2.html#K3">/.</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/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</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>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msafree2.html#E1:20_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="msualg_2.html#D8">MSUALG_2:def 8</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="msualg_2.html#K3">/.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="msafree2.html#E2:20_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="msafree2.html#E3:20_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="msualg_2.html#D8">MSUALG_2:def 8</a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:20_1"/>
the <a href="msualg_1.html#U5">Charact</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <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>7</sub></font>
 <b>by </b><i><a class="txt" href="msafree2.html#E6:20_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="msualg_2.html#D9">MSUALG_2:def 9</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font> as  <a href="msualg_1.html#V4">strict</a> <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> <b>by </b><i><a class="txt" href="msafree2.html#E4:20"><i><font color="Green">E8</font></i></a>, <a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E7:20"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D18">MSUALG_2:def 18</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E1:20_2"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E2:20_2"/><i><font color="Green">E11</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>8</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>2</sub></font>,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="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</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="E1:20_2_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/><a NAME="E3:20_2_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>10</sub></font> is <a href="msualg_2.html#V3">opers_closed</a>
 <b>by </b><i><a class="txt" href="msafree2.html#E1:20_2_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/><div><i><font color="Green">E14</font></i>: <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>11</sub></font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:20_2_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="msualg_2.html#R1">is_closed_on</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="msafree2.html#E3:20_2_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="msualg_2.html#D7">MSUALG_2:def 7</a></i>;<br/><a NAME="E2:20_2_1_1"/><b>then </b><i><font color="Green">E15</font></i>: 
 <a href="relset_1.html#K5">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>11</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>10</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>11</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</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>11</sub></font>
 <b>by </b><i><a class="ref" href="msualg_2.html#D6">MSUALG_2:def 6</a></i>;<br/> <a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>11</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>2</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>2</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>2</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>

<br/>.= 
 <a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>2</sub></font>

;<br/><b>hence </b><a NAME="E4:20_2_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="msualg_2.html#R1">is_closed_on</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="msafree2.html#E2:20_2_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="msualg_2.html#D6">MSUALG_2:def 6</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:20_2_1"/>
<font color="Maroon">c<sub>9</sub></font> is <a href="msualg_2.html#V3">opers_closed</a>
 <b>by </b><i><a class="ref" href="msualg_2.html#D7">MSUALG_2:def 7</a></i>;<br/><a NAME="E6:20_2_1"/><i><font color="Green">E15</font></i>: 
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>8</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>10</sub></font>
 <b>by </b><i><a class="txt" href="msafree2.html#E1:20_2_1"><i><font color="Green">E12</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> =  <a href="msualg_2.html#K4">Opers</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font> as    <a href="pboole.html#M3">ManySortedFunction</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</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>,<font color="Maroon">c<sub>10</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> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:20_2_1_2"/><i><font color="Green">E16</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>12</sub></font>
 <b>by </b><i><a class="txt" href="msafree2.html#E4:20_2_1"><i><font color="Green">E14</font></i></a></i>;<br/><a NAME="E2:20_2_1_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>10</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="msafree2.html#E3:20_2_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="msualg_2.html#D7">MSUALG_2:def 7</a></i>;<br/><i><font color="Green">E18</font></i>:  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>2</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>2</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>

<br/>.= 
 <a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>12</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>2</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>2</sub></font> #)

;<br/><b>thus </b><font color="Maroon">c<sub>11</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> = 
<font color="Maroon">c<sub>12</sub></font> <a href="msualg_2.html#K3">/.</a> <font color="Maroon">c<sub>9</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>12</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K8">|</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 <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="msafree2.html#E1:20_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="msafree2.html#E3:20_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="msualg_2.html#D8">MSUALG_2:def 8</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="msualg_2.html#K3">/.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="msafree2.html#E2:20_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="msualg_2.html#D8">MSUALG_2:def 8</a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:20_2_1"/>
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>8</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>9</sub></font>
 <b>by </b><i><a class="txt" href="msafree2.html#E6:20_2_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="msualg_2.html#D9">MSUALG_2:def 9</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="msafree2.html#E2:20_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/><a NAME="E5:20_2"/>
<font color="Maroon">c<sub>7</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="msafree2.html#E1:20_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="msualg_2.html#D18">MSUALG_2:def 18</a></i>;<br/><b>hence </b><a NAME="E6:20_2"/>
<font color="Maroon">c<sub>7</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>8</sub></font>
 ;<br/></div><b>end;</b></div>
<a NAME="E9:20"/><b>then </b>
 <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="msafree2.html#E7:20"><i><font color="Green">E9</font></i></a>, <a class="ref" href="msualg_2.html#D18">MSUALG_2:def 18</a></i>;<br/>
<a NAME="E10:20"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#T22">MSUALG_2:22</a></i>;<br/>
<b>hence </b><a NAME="E11:20"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> is    <a href="msafree.html#M1">GeneratorSet</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="msafree.html#D4">MSAFREE:def 4</a></i>;<br/>


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