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

<b>set </b><font color="Maroon">c<sub>1</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">F<sub>2</sub></font>() : ex <font color="Olive">b<sub>1</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>2</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">F<sub>2</sub></font>();<br/>
<a NAME="E2:11_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">F<sub>2</sub></font>()
 
<b>by </b><i><a class="ref" href="msualg_2.html#D20">MSUALG_2:def 20</a></i>;<br/>
<a NAME="E3:11_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">F<sub>2</sub></font>() : ex <font color="Olive">b<sub>1</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">F<sub>2</sub></font>() : ex <font color="Olive">b<sub>1</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<a NAME="E5:11_1"/><i><font color="Green">E8</font></i>: 
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> <font color="Maroon">c<sub>3</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E7:11_1"/><i><font color="Green">E9</font></i>: 
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> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>(<a class="txt" href="birkhoff.html#E5:11_1"><i><font color="Green">E8</font></i></a>);<br/></i>
<a NAME="E8:11_1"/>
<font color="Maroon">c<sub>4</sub></font> is    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pralg_2.html#D12">PRALG_2:def 12</a><br/>

<b>assume </b><a NAME="E1:11_1_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E3:11_1_1"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>
<b>thus </b><a NAME="E4:11_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> is  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="birkhoff.html#E7:11_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="birkhoff.html#E1:11_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="birkhoff.html#E3:11_1_1"><i><font color="Green">E11</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">F<sub>1</sub></font>() ;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E11:11_1"/><i><font color="Green">E10</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">F<sub>2</sub></font>() st <font color="Olive">b<sub>1</sub></font> <a href="msualg_3.html#R2">is_epimorphism</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="txt" href="birkhoff.html#E7:11_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="equation.html#T29">EQUATION:29</a></i>;<br/>
<a NAME="E12:11_1"/>
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> <font color="Maroon">c<sub>3</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:11_1_2"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E3:11_1_2"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E5:11_1_2"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E3:11_1_2"><i><font color="Green">E12</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>thus </b><a NAME="E6:11_1_2"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font>] )
 <b>by </b><i><a class="txt" href="birkhoff.html#E1:11"><i><font color="Green">E3</font></i></a>, <a class="txt" href="birkhoff.html#E7:11_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="birkhoff.html#E1:11_1_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="birkhoff.html#E3:11_1_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="birkhoff.html#E5:11_1_2"><i><font color="Green">E13</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:11_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[ <a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>5</sub></font>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E5:11"><i><font color="Green">E7</font></i></a></i>;<br/>
<a NAME="E14:11_1"/><b>then </b><i><font color="Green">E11</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E3:11"><i><font color="Green">E5</font></i></a></i>;<br/>
<a NAME="E15:11_1"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>() st <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="msualg_3.html#R6">are_isomorphic</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E2:11"><i><font color="Green">E4</font></i></a></i>;<br/>
<a NAME="E16:11_1"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Olive">b<sub>1</sub></font> st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E4:11"><i><font color="Green">E6</font></i></a></i>;<br/>
<b>thus </b><a NAME="E17:11_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>()]
 <b>from </b><i><a class="ref" href="birkhoff.html#S8" target="_self">BIRKHOFF:sch 8</a>(<a class="txt" href="birkhoff.html#E11:11_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="birkhoff.html#E14:11_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="birkhoff.html#E15:11_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="birkhoff.html#E16:11_1"><i><font color="Green">E13</font></i></a>);</i><br/>


</div>
