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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">U1</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>, <font color="Maroon">U2</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>, <font color="Maroon">U3</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">H</font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U1</font>,<font color="Maroon">U2</font>;<br/><b>let </b><font color="Maroon">H1</font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U2</font>,<font color="Maroon">U3</font>;<br/>







<b>assume </b><a NAME="E1:32"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">H</font> <a href="msualg_3.html#R4" target="_self">is_isomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font> &amp; <font color="Maroon">H1</font> <a href="msualg_3.html#R4" target="_self">is_isomorphism</a> <font color="Maroon">U2</font>,<font color="Maroon">U3</font> )
 ;<br/>

<a NAME="E2:32"/><b>then </b><i><font color="Green">E33</font></i>: 
( <font color="Maroon">H</font> <a href="msualg_3.html#R2" target="_self">is_epimorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font> &amp; <font color="Maroon">H</font> <a href="msualg_3.html#R3" target="_self">is_monomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:32"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">H1</font> <a href="msualg_3.html#R3" target="_self">is_monomorphism</a> <font color="Maroon">U2</font>,<font color="Maroon">U3</font> &amp; <font color="Maroon">H1</font> <a href="msualg_3.html#R2" target="_self">is_epimorphism</a> <font color="Maroon">U2</font>,<font color="Maroon">U3</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:32"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">H1</font> <a href="msualg_3.html#K3" target="_self">**</a> <font color="Maroon">H</font> <a href="msualg_3.html#R2" target="_self">is_epimorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U3</font>
 
<b>by </b><i><a class="ref" href="msualg_3.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E5:32"/>
<font color="Maroon">H1</font> <a href="msualg_3.html#K3" target="_self">**</a> <font color="Maroon">H</font> <a href="msualg_3.html#R3" target="_self">is_monomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U3</font>
 
<b>by </b><i><a class="ref" href="msualg_3.html#D1" target="_self">Def1</a>, , </i>;<br/>
<b>hence </b><a NAME="E6:32"/>
<font color="Maroon">H1</font> <a href="msualg_3.html#K3" target="_self">**</a> <font color="Maroon">H</font> <a href="msualg_3.html#R4" target="_self">is_isomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U3</font>
 <b>by </b><i><a class="txt" href="msualg_3.html#E1:32"><i><font color="Green">E30</font></i></a>, </i>;<br/>


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