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

<b>let </b><font color="Maroon" title="c1">S</font> be  <a href="osalg_4.html#V2" title="OSALG_4:attr.2">locally_directed</a> <a href="osalg_1.html#NM3" title="OSALG_1:NM.3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon" title="c2">U1</font>, <font color="Maroon" title="c3">U2</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c4">F</font> be   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font>;<br/>





<b>set </b><font color="Maroon" title="c5">mc</font> =  <a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font>;<br/>
<b>set </b><font color="Maroon" title="c6">qa</font> =  <a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">qh</font> =  <a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font>;<br/>
<b>assume </b><a NAME="E1:57"/><i><font color="Green" title="E34">A1</font></i>: 
( <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font> &amp; <font color="Maroon" title="c4">F</font> is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a> )
 ;<br/>

<a NAME="E2:57"/><b>then </b><i><font color="Green" title="E35">A2</font></i>: 
( <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c3">U2</font> &amp; <font color="Maroon" title="c4">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> )
 
<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="E3:57"/><b>then </b>
 <a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#R3" title="MSUALG_3:pred.3">is_monomorphism</a>  <a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c3">U2</font>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E1:57"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="osalg_4.html#T18" target="_self" title="OSALG_4:th.18">Th18</a></i>;<br/>
<a NAME="E4:57"/><b>then </b><i><font color="Green" title="E36">A3</font></i>: 
(  <a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c3">U2</font> &amp;  <a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> )
 
<b>by </b><i><a class="ref" href="msualg_3.html#D11" title="MSUALG_3:def.11">MSUALG_3:def 11</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">Sq</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c9">S1</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font>;<br/>
<b>set </b><font color="Maroon" title="c10">S2</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">U2</font>;<br/>
<a NAME="E5:57"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> holds <br/> <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>set </b><font color="Maroon" title="c12">f</font> = <span class="p1">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font>;<br/>
<b>assume </b><a NAME="E1:57_1"/>
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c13">s</font> = <font color="Maroon" title="c11">i</font> as   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<a NAME="E3:57_1"/><i><font color="Green" title="E37">A4</font></i>: 
<span class="p1">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_4.html#K20" title="OSALG_4:func.20">OSHomQuot</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c13">s</font>
 
<b>by </b><i><a class="ref" href="osalg_4.html#D27" target="_self" title="OSALG_4:def.27">Def27</a></i>;<br/>
<a NAME="E4:57_1"/><b>then </b><i><font color="Green" title="E38">A5</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<b>thus </b><a NAME="E5:57_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font>
 <b>by </b><i><a class="txt" href="osalg_4.html#E3:57_1"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<a NAME="E6:57_1"/><i><font color="Green" title="E39">A6</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s</font>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E2:57"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="msualg_3.html#D3" title="MSUALG_3:def.3">MSUALG_3:def 3</a></i>;<br/>
<b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E7:57_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c15">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:57_1"/><i><font color="Green" title="E40">A7</font></i>: 
( <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c13">s</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font> )
 <b>by </b><i><a class="txt" href="osalg_4.html#E6:57_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">a</font> = <font color="Maroon" title="c15">a</font> as    <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">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s</font> <b>by </b><i><a class="txt" href="osalg_4.html#E9:57_1"><i><font color="Green" title="E40">A7</font></i></a></i>;<br/>
<a NAME="E11:57_1"/><i><font color="Green" title="E41">A8</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K8" title="OSALG_4:func.8">OSClass</a> <span class="p2">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c16">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font>
 
<b>by </b><i><a class="txt" href="osalg_4.html#E1:57"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="osalg_4.html#E2:57"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="osalg_4.html#E3:57_1"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="osalg_4.html#E9:57_1"><i><font color="Green" title="E40">A7</font></i></a>, <a class="ref" href="osalg_4.html#D26" target="_self" title="OSALG_4:def.26">Def26</a></i>;<br/>
<a NAME="E12:57_1"/>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_4.html#K6" title="OSALG_4:func.6">OSClass</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c13">s</font>
 
<b>by </b><i><a class="ref" href="osalg_4.html#D13" target="_self" title="OSALG_4:def.13">Def13</a></i>;<br/>
<b>hence </b><a NAME="E13:57_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="osalg_4.html#E4:57_1"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="osalg_4.html#E11:57_1"><i><font color="Green" title="E41">A8</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:57"/><b>then </b>
 <a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a>
 
<b>by </b><i><a class="ref" href="msualg_3.html#D3" title="MSUALG_3:def.3">MSUALG_3:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:57"/>
 <a href="osalg_4.html#K21" title="OSALG_4:func.21">OSHomQuot</a> <font color="Maroon" title="c4">F</font> <a href="msualg_3.html#R4" title="MSUALG_3:pred.4">is_isomorphism</a>  <a href="osalg_4.html#K16" title="OSALG_4:func.16">QuotOSAlg</a> <font color="Maroon" title="c2">U1</font>,<span class="p1">(<span class="default"><a href="osalg_4.html#K19" title="OSALG_4:func.19">OSCng</a> <font color="Maroon" title="c4">F</font></span>)</span>,<font color="Maroon" title="c3">U2</font>
 <b>by </b><i><a class="txt" href="osalg_4.html#E4:57"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="msualg_3.html#T13" title="MSUALG_3:th.13">MSUALG_3:13</a></i>;<br/>


</div>
