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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>, <font color="Maroon" title="c3">U2</font>, <font color="Maroon" title="c4">U3</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="c5">H</font> be   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font>;<br/><b>let </b><font color="Maroon" title="c6">H1</font> be   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c3">U2</font>,<font color="Maroon" title="c4">U3</font>;<br/>







<b>assume </b><a NAME="E1:32"/><i><font color="Green" title="E27">A1</font></i>: 
( <font color="Maroon" title="c5">H</font> <a href="msualg_3.html#R4" title="MSUALG_3:pred.4">is_isomorphism</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font> &amp; <font color="Maroon" title="c6">H1</font> <a href="msualg_3.html#R4" title="MSUALG_3:pred.4">is_isomorphism</a> <font color="Maroon" title="c3">U2</font>,<font color="Maroon" title="c4">U3</font> )
 ;<br/>

<a NAME="E2:32"/><b>then </b><i><font color="Green" title="E28">A2</font></i>: 
( <font color="Maroon" title="c5">H</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font> &amp; <font color="Maroon" title="c5">H</font> <a href="msualg_3.html#R3" title="MSUALG_3:pred.3">is_monomorphism</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font> )
 
<b>by </b><i><a class="ref" href="msualg_3.html#D12" target="_self" title="MSUALG_3:def.12">Def12</a></i>;<br/>
<a NAME="E3:32"/><i><font color="Green" title="E29">A3</font></i>: 
( <font color="Maroon" title="c6">H1</font> <a href="msualg_3.html#R3" title="MSUALG_3:pred.3">is_monomorphism</a> <font color="Maroon" title="c3">U2</font>,<font color="Maroon" title="c4">U3</font> &amp; <font color="Maroon" title="c6">H1</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon" title="c3">U2</font>,<font color="Maroon" title="c4">U3</font> )
 
<b>by </b><i><a class="txt" href="msualg_3.html#E1:32"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="msualg_3.html#D12" target="_self" title="MSUALG_3:def.12">Def12</a></i>;<br/>
<a NAME="E4:32"/><b>then </b><i><font color="Green" title="E30">A4</font></i>: 
<font color="Maroon" title="c6">H1</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c5">H</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c4">U3</font>
 
<b>by </b><i><a class="txt" href="msualg_3.html#E2:32"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="msualg_3.html#T11" target="_self" title="MSUALG_3:th.11">Th11</a></i>;<br/>
<a NAME="E5:32"/>
<font color="Maroon" title="c6">H1</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c5">H</font> <a href="msualg_3.html#R3" title="MSUALG_3:pred.3">is_monomorphism</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c4">U3</font>
 
<b>by </b><i><a class="txt" href="msualg_3.html#E2:32"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="msualg_3.html#E3:32"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="msualg_3.html#T12" target="_self" title="MSUALG_3:th.12">Th12</a></i>;<br/>
<b>hence </b><a NAME="E6:32"/>
<font color="Maroon" title="c6">H1</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c5">H</font> <a href="msualg_3.html#R4" title="MSUALG_3:pred.4">is_isomorphism</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c4">U3</font>
 <b>by </b><i><a class="txt" href="msualg_3.html#E4:32"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="msualg_3.html#D12" target="_self" title="MSUALG_3:def.12">Def12</a></i>;<br/>


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