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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#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">U2</font> be    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U1</font>,<font color="Maroon">U2</font>;<br/>







<b>thus </b><a NAME="E1:29"/>
( <font color="Maroon">F</font> <a href="msualg_3.html#R4" target="_self">is_isomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font> implies ( <font color="Maroon">F</font> <a href="msualg_3.html#R1" target="_self">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font> &amp; <font color="Maroon">F</font> is <a href="msualg_3.html#V2" target="_self">"onto"</a> &amp; <font color="Maroon">F</font> is <a href="msualg_3.html#V1" target="_self">"1-1"</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1"/>
<font color="Maroon">F</font> <a href="msualg_3.html#R4" target="_self">is_isomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font>
 ;<br/>

<a NAME="E2:29_1"/><b>then </b>
( <font color="Maroon">F</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">F</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/>
<b>hence </b><a NAME="E3:29_1"/>
( <font color="Maroon">F</font> <a href="msualg_3.html#R1" target="_self">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font> &amp; <font color="Maroon">F</font> is <a href="msualg_3.html#V2" target="_self">"onto"</a> &amp; <font color="Maroon">F</font> is <a href="msualg_3.html#V1" target="_self">"1-1"</a> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:29"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">F</font> <a href="msualg_3.html#R1" target="_self">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font> &amp; <font color="Maroon">F</font> is <a href="msualg_3.html#V2" target="_self">"onto"</a> &amp; <font color="Maroon">F</font> is <a href="msualg_3.html#V1" target="_self">"1-1"</a> )
 ;<br/>

<a NAME="E3:29"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">F</font> <a href="msualg_3.html#R2" target="_self">is_epimorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:29"/>
<font color="Maroon">F</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>, </i>;<br/>
<b>hence </b><a NAME="E5:29"/>
<font color="Maroon">F</font> <a href="msualg_3.html#R4" target="_self">is_isomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font>
 <b>by </b><i>, </i>;<br/>


</div>
