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

<b>set </b><font color="Maroon" title="c1">I</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>set </b><font color="Maroon" title="c2">sA</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>();<br/>
<b>consider </b><font color="Maroon" title="c3">Gen</font> being    <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E2:9_1"/><i><font color="Green" title="E8">A6</font></i>: 
<font color="Maroon" title="c3">Gen</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a>
 <b>by </b><i><a class="ref" href="msafree2.html#D10" title="MSAFREE2:def.10">MSAFREE2:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">Gen</font> = <font color="Maroon" title="c3">Gen</font> as  <a href="finset_1.html#V2" title="FINSET_1:attr.2">V24</a>  <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 <font color="Maroon">F<sub>2</sub></font>() <b>by </b><i><a class="txt" href="birkhoff.html#E2:9_1"><i><font color="Green" title="E8">A6</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">GEN</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a> <a href="finset_1.html#V2" title="FINSET_1:attr.2">V24</a>  <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 <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E5:9_1"/><i><font color="Green" title="E9">A7</font></i>: 
<font color="Maroon" title="c4">Gen</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c5">GEN</font>
 <b>by </b><i><a class="ref" href="msualg_9.html#T8" title="MSUALG_9:th.8">MSUALG_9:8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">F</font> being    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c5">GEN</font><b> such that </b><br/><a NAME="E7:9_1"/><i><font color="Green" title="E10">A8</font></i>: 
<font color="Maroon" title="c6">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_9.html#T13" title="MSUALG_9:th.13">MSUALG_9:13</a></i>;<br/>
<a NAME="E8:9_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="pzfmisc1.html#R1" title="PZFMISC1:pred.1">is_transformable_to</a> <font color="Maroon" title="c5">GEN</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">i</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="pzfmisc1.html#D3" title="PZFMISC1:def.3">PZFMISC1:def 3</a><br/>

<b>assume </b><a NAME="E1:9_1_1"/>
<font color="Maroon" title="c7">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">F<sub>1</sub></font>()
 ;<br/>

<b>hence </b><a NAME="E2:9_1_1"/>
(  not <font color="Maroon" title="c5">GEN</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="ref" href="pboole.html#D16" title="PBOOLE:def.16">PBOOLE:def 16</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">G</font> = <font color="Maroon" title="c6">F</font> as    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <b>by </b><i><a class="ref" href="extens_1.html#T9" title="EXTENS_1:th.9">EXTENS_1:9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">h</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E11:9_1"/><i><font color="Green" title="E11">A9</font></i>: 
<font color="Maroon" title="c8">h</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()
 <b>and </b><br/><a NAME="E12:9_1"/><i><font color="Green" title="E12">A10</font></i>: 
<font color="Maroon" title="c7">G</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c8">h</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon">F<sub>4</sub></font>()
 <b>by </b><i><a class="txt" href="birkhoff.html#E1:9"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="birkhoff.html#E3:9"><i><font color="Green" title="E5">A3</font></i></a></i>;<br/>
<a NAME="E13:9_1"/><i><font color="Green" title="E13">A11</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon">F<sub>3</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c8">h</font></span>)</span>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E2:9"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="birkhoff.html#E5:9"><i><font color="Green" title="E7">A5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">sI</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="c8">h</font></span>)</span> as   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon">F<sub>2</sub></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="E15:9_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="c8">h</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="c8">h</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:9_1_2"/>
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="c8">h</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="c8">h</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="c10">sI1</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="c8">h</font></span>)</span> as   <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="c8">h</font></span>)</span> ;<br/>
<a NAME="E17:9_1"/>
<font color="Maroon" title="c5">GEN</font> is    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of <font color="Maroon" title="c9">sI</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_3"><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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a>,<a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a><br/>

<b>assume </b><a NAME="E1:9_1_3"/>
<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">F<sub>1</sub></font>()
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c12">s</font> = <font color="Maroon" title="c11">i</font> as   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<a NAME="E3:9_1_3"/><i><font color="Green" title="E14">A12</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="c6">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">GEN</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E7:9_1"><i><font color="Green" title="E10">A8</font></i></a>, <a class="ref" href="msualg_3.html#D3" title="MSUALG_3:def.3">MSUALG_3:def 3</a></i>;<br/>
<a NAME="E4:9_1_3"/><i><font color="Green" title="E15">A13</font></i>: 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">fi</font> = <font color="Maroon">F<sub>4</sub></font>() <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c12">s</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font> ;<br/>
<b>let </b><font color="Maroon" title="c14">g</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="E6:9_1_3"/>
<font color="Maroon" title="c14">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">GEN</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">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:9_1_3"/><i><font color="Green" title="E16">A14</font></i>: 
<font color="Maroon" title="c15">x</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="c6">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c12">s</font></span>)</span>
 <b>and </b><br/><a NAME="E9:9_1_3"/><i><font color="Green" title="E17">A15</font></i>: 
<font color="Maroon" title="c14">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E3:9_1_3"><i><font color="Green" title="E14">A12</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c12">s</font></span>)</span> = 
 <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
<b>by </b><i><a class="txt" href="birkhoff.html#E4:9_1_3"><i><font color="Green" title="E15">A13</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">fi</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E11:9_1_3"/><b>then </b><i><font color="Green" title="E18">A16</font></i>: 
<font color="Maroon" title="c13">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">fi</font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E8:9_1_3"><i><font color="Green" title="E16">A14</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">fi</font></span>)</span> = 
 <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">fi</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E13:9_1_3"/><b>then </b><i><font color="Green" title="E19">A17</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">fi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c12">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T27" title="FUNCT_1:th.27">FUNCT_1:27</a></i>;<br/>
<font color="Maroon" title="c14">g</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">fi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font>
<b>by </b><i><a class="txt" href="birkhoff.html#E12:9_1"><i><font color="Green" title="E12">A10</font></i></a>, <a class="txt" href="birkhoff.html#E9:9_1_3"><i><font color="Green" title="E17">A15</font></i></a>, <a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span>
<b>by </b><i><a class="txt" href="birkhoff.html#E8:9_1_3"><i><font color="Green" title="E16">A14</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
;<br/>
<a NAME="E15:9_1_3"/><b>then </b>
<font color="Maroon" title="c14">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font></span>)</span>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E11:9_1_3"><i><font color="Green" title="E18">A16</font></i></a>, <a class="txt" href="birkhoff.html#E13:9_1_3"><i><font color="Green" title="E19">A17</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E16:9_1_3"/><b>then </b>
<font color="Maroon" title="c14">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">h</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">F<sub>3</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D25" title="PBOOLE:def.25">PBOOLE:def 25</a></i>;<br/>
<b>hence </b><a NAME="E17:9_1_3"/>
<font color="Maroon" title="c14">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">sI</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="birkhoff.html#E11:9_1"><i><font color="Green" title="E11">A9</font></i></a>, <a class="ref" href="msualg_3.html#D14" title="MSUALG_3:def.14">MSUALG_3:def 14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:9_1"/><b>then </b><i><font color="Green" title="E14">A18</font></i>: 
 <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c5">GEN</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="c9">sI</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="E19:9_1"/>
 <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c9">sI</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="c10">sI1</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="E20:9_1"/><b>then </b><i><font color="Green" title="E15">A19</font></i>: 
 <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c5">GEN</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="c8">h</font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E18:9_1"><i><font color="Green" title="E14">A18</font></i></a>, <a class="ref" href="msualg_2.html#T22" title="MSUALG_2:th.22">MSUALG_2:22</a></i>;<br/>
<a NAME="E21:9_1"/>
<font color="Maroon" title="c5">GEN</font> is    <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon">F<sub>2</sub></font>()
 
<b>by </b><i><a class="txt" href="birkhoff.html#E5:9_1"><i><font color="Green" title="E9">A7</font></i></a>, <a class="ref" href="msscyc_1.html#T21" title="MSSCYC_1:th.21">MSSCYC_1:21</a></i>;<br/>
<a NAME="E22:9_1"/><b>then </b>
<font color="Maroon">F<sub>2</sub></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="c8">h</font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E20:9_1"><i><font color="Green" title="E15">A19</font></i></a>, <a class="ref" href="msafree.html#T3" title="MSAFREE:th.3">MSAFREE:3</a></i>;<br/>
<a NAME="E23:9_1"/><b>then </b>
<font color="Maroon">F<sub>2</sub></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="c8">h</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#T8" title="MSUALG_2:th.8">MSUALG_2:8</a></i>;<br/>
<a NAME="E24:9_1"/><b>then </b>
<font color="Maroon" title="c8">h</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()
 
<b>by </b><i><a class="txt" href="birkhoff.html#E11:9_1"><i><font color="Green" title="E11">A9</font></i></a>, <a class="ref" href="msualg_3.html#T19" title="MSUALG_3:th.19">MSUALG_3:19</a></i>;<br/>
<a NAME="E25:9_1"/><b>then </b>
 <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon">F<sub>3</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c8">h</font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>() <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="msualg_4.html#T6" title="MSUALG_4:th.6">MSUALG_4:6</a></i>;<br/>
<b>hence </b><a NAME="E26:9_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>()]
 <b>by </b><i><a class="txt" href="birkhoff.html#E4:9"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="birkhoff.html#E13:9_1"><i><font color="Green" title="E13">A11</font></i></a></i>;<br/>


</div>
