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

<b>set </b><font color="Maroon">c<sub>1</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>();<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="msafree.html#M1">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">E8</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_circ.html#V1">locally-finite</a>
 <b>by </b><i><a class="ref" href="msafree2.html#D10">MSAFREE2:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  <a href="pre_circ.html#V1">locally-finite</a>  <a href="pboole.html#M4">ManySortedSubset</a> of the <a href="msualg_1.html#U4">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">E8</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being  <a href="relat_1.html#V2">V5</a> <a href="pre_circ.html#V1">locally-finite</a>  <a href="pboole.html#M4">ManySortedSubset</a> of the <a href="msualg_1.html#U4">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">E9</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="msualg_9.html#T8">MSUALG_9:8</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="pboole.html#M3">ManySortedFunction</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2">--&gt;</a> <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E7:9_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="msualg_3.html#V2">"onto"</a>
 <b>by </b><i><a class="ref" href="msualg_9.html#T13">MSUALG_9:13</a></i>;<br/>
<a NAME="E8:9_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2">--&gt;</a> <a href="numbers.html#K5">NAT</a>  <a href="pzfmisc1.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>5</sub></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">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pzfmisc1.html#D3">PZFMISC1:def 3</a><br/>

<b>assume </b><a NAME="E1:9_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">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">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2">--&gt;</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="pboole.html#M3">ManySortedFunction</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2">--&gt;</a> <a href="numbers.html#K5">NAT</a> ,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <b>by </b><i><a class="ref" href="extens_1.html#T9">EXTENS_1:9</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">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">E11</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#R1">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">E12</font></i>: 
<font color="Maroon">c<sub>7</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">F<sub>4</sub></font>()
 <b>by </b><i><a class="txt" href="birkhoff.html#E1:9"><i><font color="Green">E3</font></i></a>, <a class="txt" href="birkhoff.html#E3:9"><i><font color="Green">E5</font></i></a></i>;<br/>
<a NAME="E13:9_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>3</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E2:9"><i><font color="Green">E4</font></i></a>, <a class="txt" href="birkhoff.html#E5:9"><i><font color="Green">E7</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> as   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">F<sub>2</sub></font>() <b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E15:9_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> is    <a href="pboole.html#M4">ManySortedSubset</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>8</sub></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">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> as   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> ;<br/>
<a NAME="E17:9_1"/>
<font color="Maroon">c<sub>5</sub></font> is    <a href="pboole.html#M4">ManySortedSubset</a> of <font color="Maroon">c<sub>9</sub></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">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a>,<a class="ref" href="pboole.html#D23">PBOOLE:def 23</a><br/>

<b>assume </b><a NAME="E1:9_1_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<a NAME="E3:9_1_3"/><i><font color="Green">E14</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <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>12</sub></font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E7:9_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="msualg_3.html#D3">MSUALG_3:def 3</a></i>;<br/>
<a NAME="E4:9_1_3"/><i><font color="Green">E15</font></i>: 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2">--&gt;</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">F<sub>4</sub></font>() <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> ;<br/>
<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E6:9_1_3"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:9_1_3"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E9:9_1_3"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E3:9_1_3"><i><font color="Green">E14</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> = 
 <a href="numbers.html#K5">NAT</a> 
<b>by </b><i><a class="txt" href="birkhoff.html#E4:9_1_3"><i><font color="Green">E15</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E11:9_1_3"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E8:9_1_3"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> = 
 <a href="numbers.html#K5">NAT</a> 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E13:9_1_3"/><b>then </b><i><font color="Green">E19</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T27">FUNCT_1:27</a></i>;<br/>
<font color="Maroon">c<sub>14</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="birkhoff.html#E12:9_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="birkhoff.html#E9:9_1_3"><i><font color="Green">E17</font></i></a>, <a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="birkhoff.html#E8:9_1_3"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/>
<a NAME="E15:9_1_3"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E11:9_1_3"><i><font color="Green">E18</font></i></a>, <a class="txt" href="birkhoff.html#E13:9_1_3"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E16:9_1_3"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pboole.html#K13">.:.:</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D25">PBOOLE:def 25</a></i>;<br/>
<b>hence </b><a NAME="E17:9_1_3"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E11:9_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="msualg_3.html#D14">MSUALG_3:def 14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:9_1"/><b>then </b><i><font color="Green">E14</font></i>: 
 <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>5</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="extens_1.html#T21">EXTENS_1:21</a></i>;<br/>
<a NAME="E19:9_1"/>
 <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="extens_1.html#T22">EXTENS_1:22</a></i>;<br/>
<a NAME="E20:9_1"/><b>then </b><i><font color="Green">E15</font></i>: 
 <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>5</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E18:9_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="msualg_2.html#T22">MSUALG_2:22</a></i>;<br/>
<a NAME="E21:9_1"/>
<font color="Maroon">c<sub>5</sub></font> is    <a href="msafree.html#M1">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">E9</font></i></a>, <a class="ref" href="msscyc_1.html#T21">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">MSSubAlgebra</a> of  <a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E20:9_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="msafree.html#T3">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">=</a>  <a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#T8">MSUALG_2:8</a></i>;<br/>
<a NAME="E24:9_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#R2">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">E11</font></i></a>, <a class="ref" href="msualg_3.html#T19">MSUALG_3:19</a></i>;<br/>
<a NAME="E25:9_1"/><b>then </b>
 <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>3</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">F<sub>2</sub></font>() <a href="msualg_3.html#R6">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="msualg_4.html#T6">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">E6</font></i></a>, <a class="txt" href="birkhoff.html#E13:9_1"><i><font color="Green">E13</font></i></a></i>;<br/>


</div>
