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<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>assume </b><a NAME="E1:11"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is    <a href="unialg_2.html#M3">SubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <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>;<br/>

<b>assume </b><a NAME="E2:11"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<a NAME="E3:11"/><i><font color="Green">E10</font></i>: 
 <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E1:11"><i><font color="Green">E8</font></i></a>, <a class="ref" href="mssublat.html#T7" target="_self">Th7</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <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>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_2.html#D7">MSUALG_2:def 7</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M1">Element</a> of 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>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="mssublat.html#E1:11"><i><font color="Green">E8</font></i></a>, <a class="ref" href="mssublat.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E5:11"/><i><font color="Green">E11</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <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>4</sub></font>,<span class="p3">(<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="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</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="p4">(<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>5</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E6:11"/><i><font color="Green">E12</font></i>: 
 <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>4</sub></font>,<span class="p3">(<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="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</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="p4">(<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>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <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>4</sub></font>,<span class="p3">(<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="p2">(<span class="default"><a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mssublat.html#E2:11"><i><font color="Green">E9</font></i></a>, <a class="txt" href="mssublat.html#E3:11"><i><font color="Green">E10</font></i></a>, <a class="ref" href="msualg_1.html#D9">MSUALG_1:def 9</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</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>5</sub></font>;<br/>
<a NAME="E7:11"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</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>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p2">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E1:11"><i><font color="Green">E8</font></i></a>, <a class="txt" href="mssublat.html#E2:11"><i><font color="Green">E9</font></i></a>, <a class="ref" href="mssublat.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E8:11"/><b>then </b><i><font color="Green">E13</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</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>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D10">MSUALG_1:def 10</a></i>;<br/>
<a NAME="E9:11"/>
 <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>4</sub></font>,<span class="p3">(<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="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</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="p4">(<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>5</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="mssublat.html#E1:11"><i><font color="Green">E8</font></i></a>, <a class="txt" href="mssublat.html#E2:11"><i><font color="Green">E9</font></i></a>, <a class="txt" href="mssublat.html#E5:11"><i><font color="Green">E11</font></i></a>, <a class="txt" href="mssublat.html#E6:11"><i><font color="Green">E12</font></i></a>, <a class="ref" href="mssublat.html#T8" target="_self">Th8</a></i>;<br/>
<b>hence </b><a NAME="E10:11"/>
<font color="Maroon">c<sub>3</sub></font> <a href="msualg_2.html#R1">is_closed_on</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="mssublat.html#E8:11"><i><font color="Green">E13</font></i></a>, <a class="ref" href="msualg_2.html#D6">MSUALG_2:def 6</a></i>;<br/>


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