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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>





<b>assume </b><a NAME="E1:25"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#R3">is_monomorphism</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:25"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_3.html#D11">MSUALG_3:def 11</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>4</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:25"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E5:25"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="msualg_3.html#R2">is_epimorphism</a> <font color="Maroon">c<sub>2</sub></font>, <a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="msualg_3.html#T21">MSUALG_3:21</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D13">MSUALG_3:def 13</a><br/>

<b>thus </b><a NAME="E6:25"/>
<font color="Maroon">c<sub>5</sub></font> <a href="msualg_3.html#R2">is_epimorphism</a> <font color="Maroon">c<sub>2</sub></font>, <a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="msualg_9.html#E5:25"><i><font color="Green">E10</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D12">MSUALG_3:def 12</a><br/>

<b>thus </b><a NAME="E7:25"/>
<font color="Maroon">c<sub>5</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">c<sub>2</sub></font>, <a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="msualg_9.html#E5:25"><i><font color="Green">E10</font></i></a>, <a class="ref" href="msualg_3.html#D10">MSUALG_3:def 10</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D11">MSUALG_3:def 11</a><br/>

<b>thus </b><a NAME="E8:25"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="msualg_3.html#V1">"1-1"</a>
 <b>by </b><i><a class="txt" href="msualg_9.html#E1:25"><i><font color="Green">E8</font></i></a>, <a class="txt" href="msualg_9.html#E4:25"><i><font color="Green">E9</font></i></a>, <a class="ref" href="msualg_3.html#D11">MSUALG_3:def 11</a></i>;<br/>


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