<?xml version="1.0"?>
<div class="add">

<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>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="msualg_4.html#K14" target="_self">QuotMSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>assume </b><a NAME="E1:37"/>
<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>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:37"/><b>then </b><i><font color="Green">E21</font></i>: 
( <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> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="msualg_3.html#V2">"onto"</a> )
 
<b>by </b><i><a class="ref" href="msualg_3.html#D10">MSUALG_3:def 10</a></i>;<br/>
<a NAME="E3:37"/><b>then </b>
 <a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#R3">is_monomorphism</a>  <a href="msualg_4.html#K14" target="_self">QuotMSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_4.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E4:37"/><b>then </b><i><font color="Green">E22</font></i>: 
(  <a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msualg_4.html#K14" target="_self">QuotMSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">c<sub>4</sub></font> is <a href="msualg_3.html#V1">"1-1"</a> )
 
<b>by </b><i><a class="ref" href="msualg_3.html#D11">MSUALG_3:def 11</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_4.html#K14" target="_self">QuotMSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E5:37"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/> <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>;<br/>
<b>assume </b><a NAME="E1:37_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:37_1"/><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_4.html#K19" target="_self">MSHomQuot</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_4.html#D22" target="_self">Def22</a></i>;<br/>
<a NAME="E4:37_1"/><b>then </b><i><font color="Green">E24</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_4.html#K14" target="_self">QuotMSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<b>thus </b><a NAME="E5:37_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="msualg_4.html#E3:37_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<a NAME="E6:37_1"/><i><font color="Green">E25</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_4.html#E2:37"><i><font color="Green">E21</font></i></a>, <a class="ref" href="msualg_3.html#D3">MSUALG_3:def 3</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E7:37_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:37_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="msualg_4.html#E6:37_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as    <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="Maroon">c<sub>13</sub></font> <b>by </b><i><a class="txt" href="msualg_4.html#E9:37_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E11:37_1"/><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="msualg_4.html#K17" target="_self">MSCng</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_4.html#E2:37"><i><font color="Green">E21</font></i></a>, <a class="txt" href="msualg_4.html#E3:37_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="msualg_4.html#E9:37_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="msualg_4.html#D21" target="_self">Def21</a></i>;<br/>
<a NAME="E12:37_1"/><i><font color="Green">E28</font></i>: 
 <a href="msualg_4.html#K17" target="_self">MSCng</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_4.html#K2" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_4.html#E2:37"><i><font color="Green">E21</font></i></a>, <a class="ref" href="msualg_4.html#D20" target="_self">Def20</a></i>;<br/>
<a NAME="E13:37_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_4.html#K14" target="_self">QuotMSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_4.html#K2" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_4.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E14:37_1"/><b>then </b>
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K17" target="_self">MSCng</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msualg_4.html#E4:37_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="msualg_4.html#E12:37_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E15:37_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="msualg_4.html#E11:37_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:37"/><b>then </b>
 <a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">c<sub>4</sub></font> is <a href="msualg_3.html#V2">"onto"</a>
 
<b>by </b><i><a class="ref" href="msualg_3.html#D3">MSUALG_3:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:37"/>
 <a href="msualg_4.html#K20" target="_self">MSHomQuot</a> <font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#R4">is_isomorphism</a>  <a href="msualg_4.html#K14" target="_self">QuotMSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="msualg_4.html#K18" target="_self">MSCng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="msualg_4.html#E4:37"><i><font color="Green">E22</font></i></a>, <a class="ref" href="msualg_3.html#T13">MSUALG_3:13</a></i>;<br/>


</div>
