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

<b>set </b><font color="Maroon">c<sub>1</sub></font> =  <a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>();<br/>
<a NAME="E1:4_1"/><i><font color="Green">E6</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#E1:4"><i><font color="Green">E3</font></i></a></i>;<br/>
<a NAME="E2:4_1"/><i><font color="Green">E7</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_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</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>[<font color="Olive">b<sub>2</sub></font>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E2:4"><i><font color="Green">E4</font></i></a></i>;<br/>
<a NAME="E3:4_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> <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">F<sub>1</sub></font>() st ( for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>4</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>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>4</sub></font>] ) ) holds <br/><font color="Maroon">P<sub>1</sub></font>[ <a href="pralg_2.html#K15">product</a> <font color="Olive">b<sub>2</sub></font>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E3:4"><i><font color="Green">E5</font></i></a></i>;<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="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">c<sub>3</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:4_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 <b>and </b><br/><a NAME="E6:4_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="msualg_3.html#R2">is_epimorphism</a>  <a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E7:4_1"/><i><font color="Green">E11</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_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<font color="Olive">b<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/>ex <font color="Olive">b<sub>3</sub></font> being  <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R6">=</a> <font color="Olive">b<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>4</sub></font> being  <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> st <font color="Olive">b<sub>4</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R6">=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="pboole.html#R6">=</a> <font color="Olive">b<sub>4</sub></font> ) )
 <b>from </b><i><a class="ref" href="birkhoff.html#S1" target="_self">BIRKHOFF:sch 1</a>(<a class="txt" href="birkhoff.html#E1:4_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="birkhoff.html#E2:4_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="birkhoff.html#E3:4_1"><i><font color="Green">E8</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="msafree.html#K1">||</a> <span class="p2">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="msualg_3.html#K4">""</a> </span>)</span> as    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>thus </b><a NAME="E9:4_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 <b>by </b><i><a class="txt" href="birkhoff.html#E5:4_1"><i><font color="Green">E9</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be  <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/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>



<b>assume </b><a NAME="E10:4_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>]
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E12:4_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E13:4_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="msafree.html#K1">||</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 <b>by </b><i><a class="ref" href="msafree.html#D5">MSAFREE:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E15:4_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E16:4_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E17:4_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="pboole.html#R6">=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E7:4_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="birkhoff.html#E10:4_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="birkhoff.html#E12:4_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>thus </b><a NAME="E18:4_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E15:4_1"><i><font color="Green">E15</font></i></a></i>;<br/>

<a NAME="E19:4_1"/>
 <a href="mssubfam.html#K2">rngs</a> <span class="p1">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">F<sub>2</sub></font>()
 
<b>by </b><i><a class="ref" href="extens_1.html#T14">EXTENS_1:14</a></i>;<br/>
<a NAME="E20:4_1"/><b>then </b><i><font color="Green">E17</font></i>: 
 <a href="msafree.html#K15">Reverse</a> <font color="Maroon">F<sub>2</sub></font>() is <a href="msualg_3.html#V2">"onto"</a>
 
<b>by </b><i><a class="ref" href="extens_1.html#T13">EXTENS_1:13</a></i>;<br/>
<a NAME="E21:4_1"/><i><font color="Green">E18</font></i>: 
 <a href="msafree.html#K15">Reverse</a> <font color="Maroon">F<sub>2</sub></font>() is <a href="msualg_3.html#V1">"1-1"</a>
 
<b>by </b><i><a class="ref" href="msualg_9.html#T12">MSUALG_9:12</a></i>;<br/>
<i><font color="Green">E19</font></i>: <font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="msafree.html#K1">||</a> <span class="p2">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="autalg_1.html#T13">AUTALG_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="msafree.html#K1">||</a> <span class="p2">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><a href="msualg_3.html#K2">id</a> <span class="p2">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="birkhoff.html#E20:4_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="birkhoff.html#E21:4_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="msualg_3.html#T5">MSUALG_3:5</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="msafree.html#K1">||</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
<b>by </b><i><a class="ref" href="msualg_3.html#T3">MSUALG_3:3</a></i>
;<br/>
<a NAME="E23:4_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="msafree.html#K1">||</a> <span class="p2">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="msafree.html#K1">||</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E16:4_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="extens_1.html#T8">EXTENS_1:8</a></i>;<br/>
<a NAME="E24:4_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E13:4_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="birkhoff.html#E22:4_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="autalg_1.html#T13">AUTALG_1:13</a></i>;<br/>
<b>hence </b><a NAME="E25:4_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E20:4_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="extens_1.html#T16">EXTENS_1:16</a></i>;<br/>

<a NAME="E26:4_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="msualg_3.html#V2">"onto"</a>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E6:4_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="msualg_3.html#D10">MSUALG_3:def 10</a></i>;<br/>
<a NAME="E27:4_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E6:4_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="msualg_3.html#D10">MSUALG_3:def 10</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>9</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>5</sub></font>;<br/>

<b>assume </b><a NAME="E28:4_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E29:4_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>3</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E27:4_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="msualg_3.html#T10">MSUALG_3:10</a></i>;<br/>
<b>assume </b><a NAME="E30:4_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E31:4_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="msafree.html#K1">||</a> <span class="p4">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="msualg_3.html#K4">""</a> </span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="msafree.html#K1">||</a> <span class="p4">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="msualg_3.html#K4">""</a> </span>)</span></span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E24:4_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="autalg_1.html#T13">AUTALG_1:13</a></i>;<br/>
<a NAME="E32:4_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="msafree.html#K1">||</a> <span class="p4">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="msualg_3.html#K4">""</a> </span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="msafree.html#K1">||</a> <span class="p4">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="msualg_3.html#K4">""</a> </span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="autalg_1.html#T13">AUTALG_1:13</a></i>;<br/>
<a NAME="E33:4_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="msafree.html#K1">||</a> <span class="p3">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p3">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span></span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="msafree.html#K1">||</a> <span class="p4">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="msualg_3.html#K4">""</a> </span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="autalg_1.html#T13">AUTALG_1:13</a></i>;<br/>
<a NAME="E34:4_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="msafree.html#K1">||</a> <span class="p3">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p3">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span></span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="msafree.html#K1">||</a> <span class="p3">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p3">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="autalg_1.html#T13">AUTALG_1:13</a></i>;<br/>
<a NAME="E35:4_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="msafree.html#K1">||</a> <span class="p3">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><a href="msualg_3.html#K2">id</a> <span class="p3">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span></span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="msafree.html#K1">||</a> <span class="p3">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p3">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E20:4_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="birkhoff.html#E21:4_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="msualg_3.html#T5">MSUALG_3:5</a></i>;<br/>
<a NAME="E36:4_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="msafree.html#K1">||</a> <span class="p3">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><a href="msualg_3.html#K2">id</a> <span class="p3">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span></span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="msafree.html#K1">||</a> <span class="p3">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><a href="msualg_3.html#K2">id</a> <span class="p3">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E20:4_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="birkhoff.html#E21:4_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="msualg_3.html#T5">MSUALG_3:5</a></i>;<br/>
<a NAME="E37:4_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="msafree.html#K1">||</a> <span class="p2">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="msafree.html#K1">||</a> <span class="p3">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><a href="msualg_3.html#K2">id</a> <span class="p3">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_3.html#T3">MSUALG_3:3</a></i>;<br/>
<a NAME="E38:4_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="msafree.html#K1">||</a> <span class="p2">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="msafree.html#K1">||</a> <span class="p2">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_3.html#T3">MSUALG_3:3</a></i>;<br/>
<a NAME="E39:4_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="msafree.html#K1">||</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="msafree.html#K1">||</a> <span class="p2">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="extens_1.html#T8">EXTENS_1:8</a></i>;<br/>
<a NAME="E40:4_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="msafree.html#K1">||</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="msafree.html#K1">||</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 
<b>by </b><i><a class="ref" href="extens_1.html#T8">EXTENS_1:8</a></i>;<br/>
<a NAME="E41:4_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E12:4_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="birkhoff.html#E16:4_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="birkhoff.html#E29:4_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="extens_1.html#T18">EXTENS_1:18</a></i>;<br/>
<b>hence </b><a NAME="E42:4_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E26:4_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="extens_1.html#T16">EXTENS_1:16</a></i>;<br/>


</div>
