<?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/>
<a NAME="E1:7_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#K13">.:.:</a> <font color="Maroon">F<sub>2</sub></font>() is <a href="relat_1.html#V2">non-empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D16">PBOOLE:def 16</a><br/>

<b>assume </b><a NAME="E1:7_1_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>2</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>3</sub></font> = <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,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>2</sub></font> <b>by </b><i><a class="txt" href="birkhoff.html#E1:7_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<a NAME="E4:7_1_1"/>
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>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="birkhoff.html#E1:7_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>
<a NAME="E5:7_1_1"/><b>then </b><i><font color="Green">E8</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:7_1_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#R2">meets</a> <font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<a NAME="E7:7_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#K13">.:.:</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E1:7_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="pboole.html#D25">PBOOLE:def 25</a></i>;<br/>
<b>hence </b><a NAME="E8:7_1_1"/>
not <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#K13">.:.:</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="birkhoff.html#E5:7_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="birkhoff.html#E6:7_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="relat_1.html#T151">RELAT_1:151</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:7_1"/>
<font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#K13">.:.:</a> <font color="Maroon">F<sub>2</sub></font>() is    <a href="pboole.html#M4">ManySortedSubset</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</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="E1:7_1_2"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>2</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>3</sub></font> = <font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,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>2</sub></font> <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<a NAME="E3:7_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#K13">.:.:</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E1:7_1_2"><i><font color="Green">E7</font></i></a>, <a class="ref" href="pboole.html#D25">PBOOLE:def 25</a></i>;<br/>
<b>hence </b><a NAME="E4:7_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#K13">.:.:</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</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>2</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#K13">.:.:</a> <font color="Maroon">F<sub>2</sub></font>() as  <a href="relat_1.html#V2">V5</a> <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">F<sub>3</sub></font>() <b>by </b><i><a class="txt" href="birkhoff.html#E1:7_1"><i><font color="Green">E6</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E4:7_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is    <a href="pboole.html#M4">ManySortedSubset</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>()
 
<b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">F<sub>3</sub></font>() <b>by </b><i><a class="ref" href="extens_1.html#T9">EXTENS_1:9</a></i>;<br/>
<a NAME="E6:7_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">F<sub>2</sub></font>() <a href="pzfmisc1.html#R1">is_transformable_to</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_3"><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="pzfmisc1.html#D3">PZFMISC1:def 3</a><br/>

<b>assume </b><a NAME="E1:7_1_3"/>
<font color="Maroon">c<sub>5</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="E2:7_1_3"/>
( the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</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>
<a NAME="E7:7_1"/>
<font color="Maroon">F<sub>2</sub></font>() <a href="pzfmisc1.html#R1">is_transformable_to</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_4"><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="pzfmisc1.html#D3">PZFMISC1:def 3</a><br/>

<b>assume </b><a NAME="E1:7_1_4"/>
<font color="Maroon">c<sub>5</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="E2:7_1_4"/>
( 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>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</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>
<a NAME="E8:7_1"/><b>then </b><i><font color="Green">E8</font></i>: 
 <a href="mssubfam.html#K1">doms</a> <font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#R6">=</a> <font color="Maroon">F<sub>2</sub></font>()
 
<b>by </b><i><a class="ref" href="mssubfam.html#T17">MSSUBFAM:17</a></i>;<br/>
<a NAME="E9:7_1"/><b>then </b>
 <a href="mssubfam.html#K2">rngs</a> <font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#R6">=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#K13">.:.:</a> <font color="Maroon">F<sub>2</sub></font>()
 
<b>by </b><i><a class="ref" href="equation.html#T14">EQUATION:14</a></i>;<br/>
<a NAME="E10:7_1"/><b>then </b>
 <a href="mssubfam.html#K2">rngs</a> <font color="Maroon">F<sub>4</sub></font>() 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_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D18">MSUALG_2:def 18</a></i>;<br/>
<a NAME="E11:7_1"/><b>then </b>
 <a href="mssubfam.html#K2">rngs</a> <font color="Maroon">F<sub>4</sub></font>() <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_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">F<sub>4</sub></font>() 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 <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="birkhoff.html#E6:7_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="birkhoff.html#E8:7_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="equation.html#T5">EQUATION:5</a></i>;<br/>
<a NAME="E13:7_1"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>2</sub></font>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E2:7"><i><font color="Green">E4</font></i></a>, <a class="txt" href="birkhoff.html#E3:7"><i><font color="Green">E5</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">F<sub>3</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E15:7_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">F<sub>3</sub></font>(), <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E16:7_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E17:7_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">F<sub>3</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">F<sub>3</sub></font>(), <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>6</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#E1:7"><i><font color="Green">E3</font></i></a></i>;<br/>
<a NAME="E18:7_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <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#E16:7_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="autalg_1.html#T13">AUTALG_1:13</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</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>3</sub></font>()<b> such that </b><br/><a NAME="E20:7_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon">c<sub>7</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>and </b><br/><a NAME="E21:7_1"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</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>3</sub></font>() st <font color="Olive">b<sub>1</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Olive">b<sub>1</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>7</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#E1:7"><i><font color="Green">E3</font></i></a>, <a class="txt" href="birkhoff.html#E2:7"><i><font color="Green">E4</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>6</sub></font> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>3</sub></font>() ;<br/>
<a NAME="E23:7_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#R3">is_monomorphism</a>  <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">F<sub>3</sub></font>()
 
<b>by </b><i><a class="ref" href="msualg_3.html#T22">MSUALG_3:22</a></i>;<br/>
<a NAME="E24:7_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">F<sub>3</sub></font>()
 
<b>by </b><i><a class="ref" href="msualg_3.html#D11">MSUALG_3:def 11</a></i>;<br/>
<a NAME="E25:7_1"/><b>then </b>
<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>3</sub></font>()
 
<b>by </b><i><a class="txt" href="birkhoff.html#E15:7_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="msualg_3.html#T10">MSUALG_3:10</a></i>;<br/>
<a NAME="E26:7_1"/><b>then </b><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E18:7_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="birkhoff.html#E21:7_1"><i><font color="Green">E12</font></i></a></i>;<br/>
<a NAME="E27:7_1"/><i><font color="Green">E14</font></i>: 
 <a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()
 
<b>by </b><i><a class="ref" href="msualg_3.html#T9">MSUALG_3:9</a></i>;<br/>
<a NAME="E28:7_1"/>
<font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_3.html#T4">MSUALG_3:4</a></i>;<br/>
<a NAME="E29:7_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="msualg_3.html#K3">**</a> <font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_3.html#T4">MSUALG_3:4</a></i>;<br/>
<a NAME="E30:7_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="pboole.html#R6">=</a>  <a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>()
 
<b>by </b><i><a class="txt" href="birkhoff.html#E21:7_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="birkhoff.html#E27:7_1"><i><font color="Green">E14</font></i></a></i>;<br/>
<a NAME="E31:7_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> is <a href="msualg_3.html#V2">"onto"</a>
 
<b>by </b><i><a class="ref" href="autalg_1.html#T24">AUTALG_1:24</a></i>;<br/>
<a NAME="E32:7_1"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="msualg_3.html#V2">"onto"</a>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E26:7_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="equation.html#T2">EQUATION:2</a></i>;<br/>
<a NAME="E33:7_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">F<sub>3</sub></font>() is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">F<sub>3</sub></font>()
 
<b>by </b><i><a class="ref" href="msualg_2.html#T6">MSUALG_2:6</a></i>;<br/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K13">.:.:</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="equation.html#T16">EQUATION:16</a></i>
<br/>.= 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>()
<b>by </b><i><a class="txt" href="birkhoff.html#E32:7_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="equation.html#T10">EQUATION:10</a></i>
;<br/>
<a NAME="E35:7_1"/><b>then </b>
 <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>()
 
<b>by </b><i><a class="txt" href="birkhoff.html#E33:7_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="msualg_2.html#T10">MSUALG_2:10</a></i>;<br/>
<b>hence </b><a NAME="E36:7_1"/>
<font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#K13">.:.:</a> <font color="Maroon">F<sub>2</sub></font>() is  <a href="relat_1.html#V2">V5</a>  <a href="msafree.html#M1">GeneratorSet</a> of <font color="Maroon">F<sub>3</sub></font>()
 <b>by </b><i><a class="ref" href="msafree.html#T3">MSAFREE:3</a></i>;<br/>


</div>
