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







<b>assume </b><a NAME="E1:45"/><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">g1</font> being    <a href="autalg_1.html#M2" title="AUTALG_1:mode.2">Element</a> of  <a href="autalg_1.html#K4" title="AUTALG_1:func.4">MSAAut</a> <font color="Maroon" title="c2">U1</font><b> such that </b><br/><a NAME="E3:45"/><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c5">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> 
 ;<br/>
<a NAME="E4:45"/><i><font color="Green" title="E32">A3</font></i>: 
( <font color="Maroon" title="c3">f</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> &amp; <font color="Maroon" title="c3">f</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> )
 
<b>by </b><i><a class="txt" href="autalg_1.html#E40"><i><font color="Green" title="E22">Lm3</font></i></a></i>;<br/>
<a NAME="E5:45"/>
<font color="Maroon" title="c5">g1</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="autalg_1.html#E1:45"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="autalg_1.html#E3:45"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="autalg_1.html#T35" target="_self" title="AUTALG_1:th.35">Th35</a></i>;<br/>
<a NAME="E6:45"/><b>then </b>
<font color="Maroon" title="c5">g1</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <span class="p1">(<span class="default"><a href="autalg_1.html#K6" title="AUTALG_1:func.6">MSAAutGroup</a> <font color="Maroon" title="c2">U1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></i>;<br/>
<a NAME="E7:45"/><b>then </b>
<font color="Maroon" title="c5">g1</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c3">f</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <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>
 
<b>by </b><i><a class="ref" href="autalg_1.html#T36" target="_self" title="AUTALG_1:th.36">Th36</a></i>;<br/>
<b>hence </b><a NAME="E8:45"/>
<font color="Maroon" title="c3">f</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> 
 <b>by </b><i><a class="txt" href="autalg_1.html#E3:45"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="autalg_1.html#E4:45"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="autalg_1.html#T19" target="_self" title="AUTALG_1:th.19">Th19</a></i>;<br/>


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