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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#V4">strict</a> <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_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>thus </b><a NAME="E1:10"/>
( <font color="Maroon">c<sub>3</sub></font> is    <a href="msafree.html#M1" target="_self">GeneratorSet</a> of <font color="Maroon">c<sub>2</sub></font> implies  <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is    <a href="msafree.html#M1" target="_self">GeneratorSet</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msualg_2.html#T6">MSUALG_2:6</a></i>;<br/>
<a NAME="E3:10_1"/>
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>3</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>4</sub></font>
 
<b>by </b><i><a class="txt" href="msafree.html#E1:10_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="msafree.html#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E4:10_1"/>
 <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="msualg_2.html#T10">MSUALG_2:10</a></i>;<br/>


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

<b>assume </b><a NAME="E2:10"/>
 <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>hence </b><a NAME="E3:10"/>
<font color="Maroon">c<sub>3</sub></font> is    <a href="msafree.html#M1" target="_self">GeneratorSet</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="msafree.html#D4" target="_self">Def4</a></i>;<br/>


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