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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1_1"/>
(  not <font color="Maroon" title="c1">S</font> is <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a> or <font color="Maroon" title="c1">S</font> is <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a> )
 </span>;<br/> <i><font color="Red">:: according to </font></i><a class="ref" href="msafree2.html#D10" title="MSAFREE2:def.10">MSAFREE2:def 10</a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_1"><b>case </b></a><a NAME="E1:25_1_1_1_1"/>
not <font color="Maroon" title="c1">S</font> is <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a>
 ;<br/><div class="add"><b>let </b><font color="Maroon" title="c4">S'</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">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>assume </b><a NAME="E2:25_1_1_1_1"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c4">S'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font>
 ;<br/><b>let </b><font color="Maroon" title="c5">H</font> be    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c4">S'</font>;<br/><b>assume </b><a NAME="E3:25_1_1_1_1"/><i><font color="Green" title="E14">A2</font></i>: 
<font color="Maroon" title="c5">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c3">X</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c6">T</font> = <font color="Maroon" title="c3">X</font> as   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c5">H</font> <b>by </b><i><a class="txt" href="equation.html#E2:25_1_1_1_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="msualg_2.html#D18" title="MSUALG_2:def.18">MSUALG_2:def 18</a></i>;<br/><a NAME="E5:25_1_1_1_1"/>
 <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c6">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H</font>
 <b>by </b><i><a class="txt" href="equation.html#E2:25_1_1_1_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="equation.html#E3:25_1_1_1_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="extens_1.html#T22" title="EXTENS_1:th.22">EXTENS_1:22</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">T</font> = <font color="Maroon" title="c6">T</font> as    <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c5">H</font> <b>by </b><i><a class="txt" href="equation.html#E2:25_1_1_1_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="equation.html#E3:25_1_1_1_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="msafree.html#T3" title="MSAFREE:th.3">MSAFREE:3</a></i>;<br/><b>take </b>
<font color="Maroon" title="c7">T</font>
;<br/><b>thus </b><a NAME="E7:25_1_1_1_1"/>
<font color="Maroon" title="c7">T</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_2"><b>case </b></a><a NAME="E1:25_1_1_1_2"/>
<font color="Maroon" title="c1">S</font> is <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:25_1_1_1_2"/>
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">X</font></span>)</span> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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