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<span class="kw">let </span><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="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon" title="c1">S</font><br/>  for <font color="Olive" title="b2">C1</font>, <font color="Olive" title="b3">C2</font> being   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Olive" title="b1">A</font><br/>  for <font color="Olive" title="b4">G</font> being   <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#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">C1</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Olive" title="b1">A</font>,<font color="Olive" title="b3">C2</font>)</span>)</span>  st (  for <font color="Olive" title="b5">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="b6">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">C1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font><br/>  for <font color="Olive" title="b7">xx</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Olive" title="b1">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font>  st <font color="Olive" title="b6">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="Olive" title="b2">C1</font>,<font color="Olive" title="b7">xx</font>) holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b4">G</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b5">i</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">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="Olive" title="b3">C2</font>,<font color="Olive" title="b7">xx</font>) ) holds <br/><font color="Olive" title="b4">G</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a>  <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">C1</font>), <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Olive" title="b1">A</font>,<font color="Olive" title="b3">C2</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon" title="c1">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">C1</font>, <font color="Olive" title="b2">C2</font> being   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font><br/>  for <font color="Olive" title="b3">G</font> being   <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#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">C1</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Maroon" title="c2">A</font>,<font color="Olive" title="b2">C2</font>)</span>)</span>  st (  for <font color="Olive" title="b4">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="b5">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">C1</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font><br/>  for <font color="Olive" title="b6">xx</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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="b4">i</font>  st <font color="Olive" title="b5">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="Olive" title="b1">C1</font>,<font color="Olive" title="b6">xx</font>) holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">G</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">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="Olive" title="b2">C2</font>,<font color="Olive" title="b6">xx</font>) ) holds <br/><font color="Olive" title="b3">G</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a>  <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">C1</font>), <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Maroon" title="c2">A</font>,<font color="Olive" title="b2">C2</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">C1</font>, <font color="Maroon" title="c4">C2</font> be   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon" title="c2">A</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G</font> being   <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#K13" title="MSUALG_4:func.13">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#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">C2</font>)</span>)</span>  st (  for <font color="Olive" title="b2">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="b3">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_4.html#K13" title="MSUALG_4:func.13">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="b2">i</font><br/>  for <font color="Olive" title="b4">xx</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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="b2">i</font>  st <font color="Olive" title="b3">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="b4">xx</font>) holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">G</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">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="b4">xx</font>) ) holds <br/><font color="Olive" title="b1">G</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a>  <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">C1</font>), <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">C2</font>)</span><br/>





<a NAME="E1:46"/>
 <a href="msualg_4.html#K15" title="MSUALG_4:func.15">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#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">C2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_4.html#T3" onmouseover="rs('msualg_4/T3')" onmouseout="rh()">MSUALG_4:3</a></span>;<br/>
<a NAME="E2:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A1</font></span>: 
 <a href="msualg_4.html#K15" title="MSUALG_4:func.15">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#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">C2</font>)
 
;<br/>
<span class="kw">let </span><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#K13" title="MSUALG_4:func.13">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#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">C2</font>)</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  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#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_4.html#K13" title="MSUALG_4:func.13">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#U3" title="MSUALG_1:sel.3">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>) ) implies <font color="Maroon" title="c5">G</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a>  <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">C1</font>), <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">C2</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E3:46"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
 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#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_4.html#K13" title="MSUALG_4:func.13">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#U3" title="MSUALG_1:sel.3">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>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">G</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a>  <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">C1</font>), <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">C2</font>)</span><br/>

<a NAME="E4:46"/>
<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#K15" title="MSUALG_4:func.15">MSNat_Hom</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">C1</font>)</span>)</span> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a>  <a href="msualg_4.html#K15" title="MSUALG_4:func.15">MSNat_Hom</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">C2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="msualg_9.html#T34" target="_self" onmouseover="rs('msualg_9/T34')" onmouseout="rh()">Th34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:46"/>
<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#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">C1</font>), <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">C2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="msualg_9.html#T19" target="_self" onmouseover="rs('msualg_9/T19')" onmouseout="rh()">Th19</a>, <a class="ref" href="msualg_4.html#T3" onmouseover="rs('msualg_4/T3')" onmouseout="rh()">MSUALG_4:3</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="msualg_3.html#D8" onmouseover="rs('msualg_3/D8')" onmouseout="rh()">MSUALG_3:def 8</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">G</font> is  <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> </span><br/>

<span class="kw">thus </span><a NAME="E6:46"/>
<font color="Maroon" title="c5">G</font> is  <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="msualg_9.html#E36"><span class="lab"><font color="Green" title="E16">Lm1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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