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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">U1</font> be  <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">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="endalg.html#K6" title="ENDALG:func.6">MSAEndMonoid</a> <font color="Maroon" title="c2">U1</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">g</font> be    <a href="autalg_1.html#M2" title="AUTALG_1:mode.2">Element</a> of  <a href="endalg.html#K4" title="ENDALG:func.4">MSAEnd</a> <font color="Maroon" title="c2">U1</font>;<br/>







<b>assume </b><a NAME="E1:22"/><i><font color="Green" title="E17">A1</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c7">i</font> =  <a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> as    <a href="autalg_1.html#M2" title="AUTALG_1:mode.2">Element</a> of  <a href="endalg.html#K4" title="ENDALG:func.4">MSAEnd</a> <font color="Maroon" title="c2">U1</font> <b>by </b><i><a class="ref" href="endalg.html#T10" target="_self" title="ENDALG:th.10">Th10</a></i>;<br/>
<a NAME="E3:22"/>
 <a href="endalg.html#K6" title="ENDALG:func.6">MSAEndMonoid</a> <font color="Maroon" title="c2">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">multLoopStr</a>(# <span class="p1">(<span class="default"><a href="endalg.html#K4" title="ENDALG:func.4">MSAEnd</a> <font color="Maroon" title="c2">U1</font></span>)</span>,<span class="p1">(<span class="default"><a href="endalg.html#K5" title="ENDALG:func.5">MSAEndComp</a> <font color="Maroon" title="c2">U1</font></span>)</span>,<font color="Maroon" title="c7">i</font> #)
 
<b>by </b><i><a class="ref" href="endalg.html#D6" target="_self" title="ENDALG:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E4:22"/>
<font color="Maroon" title="c3">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c5">f</font>
 <b>by </b><i><a class="txt" href="endalg.html#E1:22"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="endalg.html#D5" target="_self" title="ENDALG:def.5">Def5</a></i>;<br/>


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