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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">A</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">B</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">C</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">h1</font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font>;<br/>







<b>assume </b><a NAME="E1:24"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">h1</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">B</font>,<font color="Maroon">C</font>
 ;<br/>

<b>let </b><font color="Maroon">h2</font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">B</font>,<font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E2:24"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">h1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">h2</font>
 ;<br/>

<a NAME="E3:24"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">C</font> is    <a href="pboole.html#M4">ManySortedSubset</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E4:24"/><b>then </b>
 <a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">C</font> is   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">C</font>,<font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="extens_1.html#T9">EXTENS_1:9</a></i>;<br/>
<b>then consider </b><font color="Maroon">G</font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">C</font>,<font color="Maroon">A</font><b> such that </b><br/><a NAME="E6:24"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">G</font> <a href="pboole.html#R6">=</a>  <a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">C</font>
 ;<br/>
<a NAME="E7:24"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">G</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">h1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">h1</font>
 
<b>by </b><i>, <a class="ref" href="msualg_3.html#T4">MSUALG_3:4</a></i>;<br/>
<a NAME="E8:24"/>
<font color="Maroon">G</font> <a href="msualg_3.html#R3">is_monomorphism</a> <font color="Maroon">C</font>,<font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="msualg_3.html#T22">MSUALG_3:22</a></i>;<br/>
<a NAME="E9:24"/><b>then </b>
<font color="Maroon">G</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">C</font>,<font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="msualg_3.html#D11">MSUALG_3:def 11</a></i>;<br/>
<b>hence </b><a NAME="E10:24"/>
<font color="Maroon">h2</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">B</font>,<font color="Maroon">A</font>
 <b>by </b><i>, , , <a class="ref" href="msualg_3.html#T10">MSUALG_3:10</a></i>;<br/>


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