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<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon" title="c2">U0</font> be  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">U0</font>;<br/>





<b>thus </b><a NAME="E1:10"/>
( <font color="Maroon" title="c3">A</font> is    <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c2">U0</font> implies  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">U0</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" title="E6">A1</font></i>: 
<font color="Maroon" title="c3">A</font> is    <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c2">U0</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">U1</font> = <font color="Maroon" title="c2">U0</font> as    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c2">U0</font> <b>by </b><i><a class="ref" href="msualg_2.html#T6" title="MSUALG_2:th.6">MSUALG_2:6</a></i>;<br/>
<a NAME="E3:10_1"/>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c4">U1</font>
 
<b>by </b><i><a class="txt" href="msafree.html#E1:10_1"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="msafree.html#D4" target="_self" title="MSAFREE:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E4:10_1"/>
 <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">U0</font>
 <b>by </b><i><a class="ref" href="msualg_2.html#T10" title="MSUALG_2:th.10">MSUALG_2:10</a></i>;<br/>


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

<b>assume </b><a NAME="E2:10"/>
 <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">U0</font>
 ;<br/>

<b>hence </b><a NAME="E3:10"/>
<font color="Maroon" title="c3">A</font> is    <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c2">U0</font>
 <b>by </b><i><a class="ref" href="msafree.html#D4" target="_self" title="MSAFREE:def.4">Def4</a></i>;<br/>


</div>
