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<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">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">A</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="c3">C1</font>, <font color="Maroon" title="c4">C2</font> be   <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon" title="c2">A</font>;<br/><b>let </b><font color="Maroon" title="c5">G</font> be   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">C1</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">C2</font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:48"/><i><font color="Green" title="E20">A1</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">C1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font><br/> for <font color="Olive" title="b3">xx</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c3">C1</font>,<font color="Olive" title="b3">xx</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_4.html#K3" title="MSUALG_4:func.3">Class</a> <font color="Maroon" title="c4">C2</font>,<font color="Olive" title="b3">xx</font>
 ;<br/>

<a NAME="E2:48"/>
 <a href="msualg_4.html#K16" title="MSUALG_4:func.16">MSNat_Hom</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">C2</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon" title="c2">A</font>, <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">C2</font>
 
<b>by </b><i><a class="ref" href="msualg_4.html#T3" title="MSUALG_4:th.3">MSUALG_4:3</a></i>;<br/>
<a NAME="E3:48"/><b>then </b><i><font color="Green" title="E21">A2</font></i>: 
 <a href="msualg_4.html#K16" title="MSUALG_4:func.16">MSNat_Hom</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">C2</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">A</font>, <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">C2</font>
 
<b>by </b><i><a class="ref" href="msualg_3.html#D10" title="MSUALG_3:def.10">MSUALG_3:def 10</a></i>;<br/>
<a NAME="E4:48"/>
<font color="Maroon" title="c5">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K16" title="MSUALG_4:func.16">MSNat_Hom</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">C1</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_4.html#K16" title="MSUALG_4:func.16">MSNat_Hom</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">C2</font>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E1:48"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="msualg_9.html#T35" target="_self" title="MSUALG_9:th.35">Th35</a></i>;<br/>
<b>hence </b><a NAME="E5:48"/>
<font color="Maroon" title="c5">G</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">C1</font>, <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">C2</font>
 <b>by </b><i><a class="txt" href="msualg_9.html#E3:48"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="msualg_9.html#T20" target="_self" title="MSUALG_9:th.20">Th20</a>, <a class="ref" href="msualg_4.html#T3" title="MSUALG_4:th.3">MSUALG_4:3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D10" title="MSUALG_3:def.10">MSUALG_3:def 10</a><br/>

<b>thus </b><a NAME="E6:48"/>
<font color="Maroon" title="c5">G</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a>
 <b>by </b><i><a class="txt" href="msualg_9.html#E1:48"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="msualg_9.html#E37"><i><font color="Green" title="E17">Lm1</font></i></a></i>;<br/>


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