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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is  <a href="msafree2.html#V4" title="MSAFREE2:attr.4">finite-yielding</a>  implies <font color="Maroon" title="c2">A</font> is  <a href="msafree2.html#V3" title="MSAFREE2:attr.3">finitely-generated</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:25_1_1"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="msafree2.html#V4" title="MSAFREE2:attr.4">finite-yielding</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="25_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:25_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/> <span class="comment"><font color="Red">:: according to </font></span><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="25_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:25_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">S9</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">S9</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">S9</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">finite-yielding</a> </span><br/><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">S9</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">S9</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">S9</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">finite-yielding</a>  )</span><br/><span class="kw">assume </span><a NAME="E2:25_1_1_1_1"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c3">S9</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">S9</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">finite-yielding</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">A9</font> be    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c3">S9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">A9</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">A9</font> st <font color="Olive" title="b1">G</font> is  <a href="finset_1.html#V2" title="FINSET_1:attr.2">finite-yielding</a>  )</span><br/><span class="kw">assume </span><a NAME="E3:25_1_1_1_1"/>
<font color="Maroon" title="c4">A9</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">A9</font> st <font color="Olive" title="b1">G</font> is  <a href="finset_1.html#V2" title="FINSET_1:attr.2">finite-yielding</a> </span><br/><span class="kw">then </span><span class="kw">reconsider </span><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">A9</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25_1_1_1_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="msafree2.html#T9" target="_self" title="MSAFREE2:th.9">Th9</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c5">G</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">G</font> is  <a href="finset_1.html#V2" title="FINSET_1:attr.2">finite-yielding</a> </span><br/><span class="kw">thus </span><a NAME="E5:25_1_1_1_1"/>
<font color="Maroon" title="c5">G</font> is  <a href="finset_1.html#V2" title="FINSET_1:attr.2">finite-yielding</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1_1"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="msafree2.html#D11" target="_self" title="MSAFREE2:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:25_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">finite-yielding</a> </span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:25_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">finite-yielding</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1_1"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="msafree2.html#D11" target="_self" title="MSAFREE2:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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