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

<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">U0</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">B</font> be  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <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="c4">G</font> be    <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c2">U0</font>;<br/><b>let </b><font color="Maroon" title="c5">H</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a>  <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c3">B</font>;<br/><b>let </b><font color="Maroon" title="c6">f</font> be   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c3">B</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:30"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c5">H</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">f</font> <a href="pboole.html#K13" title="PBOOLE:func.13">.:.:</a> <font color="Maroon" title="c4">G</font>
 <b>and </b><br/><a NAME="E2:30"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c3">B</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c7">I</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Maroon" title="c6">f</font></span>)</span> as  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a> <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c3">B</font> <b>by </b><i><a class="ref" href="msualg_2.html#D10" title="MSUALG_2:def.10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E4:30"/>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Maroon" title="c6">f</font></span>)</span> is    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</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_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Maroon" title="c6">f</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:30_1"/>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Maroon" title="c6">f</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c8">I1</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Maroon" title="c6">f</font></span>)</span> as  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a> <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Maroon" title="c6">f</font></span>)</span> ;<br/>
<a NAME="E6:30"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c7">I</font> is    <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of  <a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="ref" href="msafree2.html#T9" title="MSAFREE2:th.9">MSAFREE2:9</a></i>;<br/>
<a NAME="E7:30"/>
 <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c7">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c8">I1</font>
 
<b>by </b><i><a class="ref" href="extens_1.html#T22" title="EXTENS_1:th.22">EXTENS_1:22</a></i>;<br/>
<a NAME="E8:30"/><b>then </b><i><font color="Green" title="E18">A4</font></i>: 
 <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c7">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="txt" href="equation.html#E6:30"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="msafree.html#T3" title="MSAFREE:th.3">MSAFREE:3</a></i>;<br/>
<a NAME="E9:30"/>
<font color="Maroon" title="c5">H</font> is    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</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_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Maroon" title="c6">f</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<a NAME="E1:30_2"/>
<font color="Maroon" title="c6">f</font> <a href="pboole.html#K13" title="PBOOLE:func.13">.:.:</a> <font color="Maroon" title="c4">G</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">f</font> <a href="pboole.html#K13" title="PBOOLE:func.13">.:.:</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U0</font>
 
<b>by </b><i><a class="ref" href="extens_1.html#T6" title="EXTENS_1:th.6">EXTENS_1:6</a></i>;<br/>
<a NAME="E2:30_2"/><b>then </b>
<font color="Maroon" title="c5">H</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">f</font> <a href="pboole.html#K13" title="PBOOLE:func.13">.:.:</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U0</font>
 
<b>by </b><i><a class="txt" href="equation.html#E1:30"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="pboole.html#T15" title="PBOOLE:th.15">PBOOLE:15</a></i>;<br/>
<b>hence </b><a NAME="E3:30_2"/>
<font color="Maroon" title="c5">H</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="equation.html#E2:30"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="msualg_3.html#D14" title="MSUALG_3:def.14">MSUALG_3:def 14</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a><br/>


</div><b>end;</b></div>
<a NAME="E10:30"/><b>then </b>
 <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c5">H</font> is    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c7">I</font>
 
<b>by </b><i><a class="ref" href="extens_1.html#T21" title="EXTENS_1:th.21">EXTENS_1:21</a></i>;<br/>
<a NAME="E11:30"/><b>then </b>
<font color="Maroon" title="c3">B</font> is    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of  <a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="txt" href="equation.html#E8:30"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="msafree.html#T3" title="MSAFREE:th.3">MSAFREE:3</a></i>;<br/>
<a NAME="E12:30"/><b>then </b>
 <a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#T8" title="MSUALG_2:th.8">MSUALG_2:8</a></i>;<br/>
<b>hence </b><a NAME="E13:30"/>
<font color="Maroon" title="c6">f</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="equation.html#E2:30"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="msualg_3.html#T19" title="MSUALG_3:th.19">MSUALG_3:19</a></i>;<br/>


</div>
