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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> 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>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:48"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_4.html#K3">Class</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>3</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_4.html#K3">Class</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font>
 ;<br/>

<a NAME="E2:48"/><i><font color="Green">E21</font></i>: 
 <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="msualg_3.html#R2">is_epimorphism</a> <font color="Maroon">c<sub>2</sub></font>, <a href="msualg_4.html#K14">QuotMSAlg</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_4.html#T3">MSUALG_4:3</a></i>;<br/>
<a NAME="E3:48"/>
 <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#R2">is_epimorphism</a> <font color="Maroon">c<sub>2</sub></font>, <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_4.html#T3">MSUALG_4:3</a></i>;<br/>
<a NAME="E4:48"/><b>then </b><i><font color="Green">E22</font></i>: 
 <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">c<sub>2</sub></font>,<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>, <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_3.html#D10">MSUALG_3:def 10</a></i>;<br/>
<a NAME="E5:48"/>
<font color="Maroon">c<sub>5</sub></font> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#R6">=</a>  <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E1:48"><i><font color="Green">E20</font></i></a>, <a class="ref" href="msualg_9.html#T35" target="_self">Th35</a></i>;<br/>
<b>hence </b><a NAME="E6:48"/>
<font color="Maroon">c<sub>5</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>, <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="msualg_9.html#E2:48"><i><font color="Green">E21</font></i></a>, <a class="txt" href="msualg_9.html#E4:48"><i><font color="Green">E22</font></i></a>, <a class="ref" href="msualg_9.html#T20" target="_self">Th20</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D10">MSUALG_3:def 10</a><br/>

<b>thus </b><a NAME="E7:48"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="msualg_3.html#V2">"onto"</a>
 <b>by </b><i><a class="txt" href="msualg_9.html#E1:48"><i><font color="Green">E20</font></i></a>, <a class="txt" href="msualg_9.html#E37"><i><font color="Green">Lemma17</font></i></a></i>;<br/>


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