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<b>let </b><font color="Maroon">S</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">A</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">X</font> be  <a href="relat_1.html#V2">V2</a>  <a href="msafree.html#M1">GeneratorSet</a> of <font color="Maroon">A</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:45"/><i><font color="Green">E34</font></i>: 
not <font color="Maroon">A</font> is <a href="msafree2.html#V4">locally-finite</a>
 <b>and </b><br/><a NAME="E2:45"/><i><font color="Green">E42</font></i>: 
 <a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font> is <a href="msafree2.html#V4">locally-finite</a>
 ;<br/>

<b>consider </b><font color="Maroon">F</font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">A</font><b> such that </b><br/><a NAME="E4:45"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">F</font> <a href="msualg_3.html#R2">is_epimorphism</a>  <a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font>,<font color="Maroon">A</font>
 <b>by </b><i/>;<br/>
<a NAME="E5:45"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">F</font> is <a href="msualg_3.html#V2">"onto"</a>
 
<b>by </b><i>, <a class="ref" href="msualg_3.html#D10">MSUALG_3:def 10</a></i>;<br/>
<a NAME="E6:45"/>
not the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> is <a href="finset_1.html#V2">locally-finite</a>
 
<b>by </b><i>, <a class="ref" href="msafree2.html#D11">MSAFREE2:def 11</a></i>;<br/>
<b>then consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:45"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; not the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i><a class="ref" href="pre_circ.html#D3">PRE_CIRC:def 3</a></i>;<br/>
<a NAME="E9:45"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> is <a href="finset_1.html#V2">locally-finite</a>
 
<b>by </b><i>, <a class="ref" href="msafree2.html#D11">MSAFREE2:def 11</a></i>;<br/>
<a NAME="E10:45"/><b>then </b><i><font color="Green">E49</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="pre_circ.html#D3">PRE_CIRC:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">SAi</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">FXi</font> = the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">Fi</font> = <font color="Maroon">F</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">i</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">FXi</font>,<font color="Maroon">SAi</font> ;<br/>
<a NAME="E15:45"/><i><font color="Green">E50</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">SAi</font>
 
<b>by </b><i>, <a class="ref" href="msualg_3.html#D3">MSUALG_3:def 3</a></i>;<br/>
<a NAME="E16:45"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">Fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FXi</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E17:45"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="finset_1.html#T26">FINSET_1:26</a></i>;<br/>


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