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<b>let </b><font color="Maroon" title="c2">A</font> be  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is <a href="msafree2.html#V4" title="MSAFREE2:attr.4">locally-finite</a> implies <font color="Maroon" title="c2">A</font> is <a href="msafree2.html#V3" title="MSAFREE2:attr.3">finitely-generated</a> )</span><br/>

<b>assume </b><a NAME="E1:23_1_1"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="msafree2.html#V4" title="MSAFREE2:attr.4">locally-finite</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">A</font> is <a href="msafree2.html#V3" title="MSAFREE2:attr.3">finitely-generated</a></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_1"/>
(  not <font color="Maroon" title="c1">S</font> is <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a> or <font color="Maroon" title="c1">S</font> is <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a> )
 </span>;<br/> <i><font color="Red">:: according to </font></i><a class="ref" href="msafree2.html#D10" target="_self" title="MSAFREE2:def.10">MSAFREE2:def 10</a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_1"><b>case </b></a><a NAME="E1:23_1_1_1_1"/>
not <font color="Maroon" title="c1">S</font> is <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S'</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>   st <font color="Olive" title="b1">S'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font> holds <br/>for <font color="Olive" title="b2">A</font> being   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Olive" title="b1">S'</font>  st <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> holds <br/> ex <font color="Olive" title="b3">G</font> being   <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Olive" title="b2">A</font> st <font color="Olive" title="b3">G</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a></span><br/><div class="add"><b>let </b><font color="Maroon" title="c3">S'</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">S'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font> implies for <font color="Olive" title="b1">A</font> being   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c3">S'</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> holds <br/> ex <font color="Olive" title="b2">G</font> being   <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Olive" title="b1">A</font> st <font color="Olive" title="b2">G</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a> )</span><br/><b>assume </b><a NAME="E2:23_1_1_1_1"/><i><font color="Green" title="E10">A2</font></i>: 
<font color="Maroon" title="c3">S'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c3">S'</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> holds <br/> ex <font color="Olive" title="b2">G</font> being   <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Olive" title="b1">A</font> st <font color="Olive" title="b2">G</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a></span><br/><b>let </b><font color="Maroon" title="c4">A'</font> be    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c3">S'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> implies  ex <font color="Olive" title="b1">G</font> being   <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c4">A'</font> st <font color="Olive" title="b1">G</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a> )</span><br/><b>assume </b><a NAME="E3:23_1_1_1_1"/>
<font color="Maroon" title="c4">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">G</font> being   <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c4">A'</font> st <font color="Olive" title="b1">G</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a></span><br/><b>then reconsider </b><font color="Maroon" title="c5">G</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> as    <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c4">A'</font> <b>by </b><i><a class="txt" href="#E2:23_1_1_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="msafree2.html#T9" target="_self" title="MSAFREE2:th.9">Th9</a></i>;<br/><b>take </b>
<font color="Maroon" title="c5">G</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">G</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a></span><br/><b>thus </b><a NAME="E5:23_1_1_1_1"/>
<font color="Maroon" title="c5">G</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a>
 <b>by </b><i><a class="txt" href="#E1:23_1_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="msafree2.html#D11" target="_self" title="MSAFREE2:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_2"><b>case </b></a><a NAME="E1:23_1_1_1_2"/>
<font color="Maroon" title="c1">S</font> is <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a></span><br/><div class="add"><b>thus </b><a NAME="E2:23_1_1_1_2"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a>
 <b>by </b><i><a class="txt" href="#E1:23_1_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="msafree2.html#D11" target="_self" title="MSAFREE2:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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