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

<a NAME="E1:14_1"/><i><font color="Green" title="E7">A5</font></i>: 
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">B</font>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E1:14"><i><font color="Green" title="E3">A1</font></i></a></i>;<br/>
<a NAME="E2:14_1"/><i><font color="Green" title="E8">A6</font></i>: 
for <font color="Olive" title="b1">A</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">B</font> being <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Olive" title="b1">A</font>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">B</font>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E2:14"><i><font color="Green" title="E4">A2</font></i></a></i>;<br/>
<a NAME="E3:14_1"/><i><font color="Green" title="E9">A7</font></i>: 
for <font color="Olive" title="b1">A</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Olive" title="b1">A</font>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">R</font>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E3:14"><i><font color="Green" title="E5">A3</font></i></a></i>;<br/>
<a NAME="E4:14_1"/><i><font color="Green" title="E10">A8</font></i>: 
for <font color="Olive" title="b1">I</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">F</font> being   <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Olive" title="b1">I</font>,<font color="Maroon">F<sub>1</sub></font>()  st ( for <font color="Olive" title="b3">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">I</font> holds <br/> ex <font color="Olive" title="b4">A</font> being   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive" title="b4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b4">A</font>] ) ) holds <br/><font color="Maroon">P<sub>1</sub></font>[ <a href="pralg_2.html#K15" title="PRALG_2:func.15">product</a> <font color="Olive" title="b2">F</font>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E4:14"><i><font color="Green" title="E6">A4</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">s</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">e</font> where <font color="Olive" title="b2">e</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4" title="EQUATION:func.4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> : for <font color="Olive" title="b3">A</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b3">A</font>] holds <br/><font color="Olive" title="b3">A</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Olive" title="b2">e</font> </span>}</span>  );<br/>
<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/>
<div><i><font color="Green" title="E11">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:14_1_1"/>
<font color="Maroon" title="c2">w</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 consider </b><font color="Maroon" title="c3">s</font> being   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:14_1_1"/><i><font color="Green" title="E12">A10</font></i>: 
<font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">w</font>
 ;<br/><b>take </b><font color="Maroon" title="c4">d</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">e</font> where <font color="Olive" title="b1">e</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4" title="EQUATION:func.4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s</font> : for <font color="Olive" title="b2">A</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">A</font>] holds <br/><font color="Olive" title="b2">A</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Olive" title="b1">e</font> </span>}</span> ;<br/><b>thus </b><a NAME="E4:14_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c4">d</font>]
 <b>by </b><i><a class="txt" href="birkhoff.html#E3:14_1_1"><i><font color="Green" title="E12">A10</font></i></a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c2">E</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</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>()<b> such that </b><br/><a NAME="E7:14_1"/><i><font color="Green" title="E12">A11</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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>() holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3" title="PBOOLE:sch.3">PBOOLE:sch 3</a>(<a class="txt" href="birkhoff.html#E5:14_1"><i><font color="Green" title="E11">A9</font></i></a>);<br/></i>
<a NAME="E8:14_1"/>
<font color="Maroon" title="c2">E</font> is    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="equation.html#K4" title="EQUATION:func.4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">j</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="E1:14_1_2"/>
<font color="Maroon" title="c3">j</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 consider </b><font color="Maroon" title="c4">s</font> being   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:14_1_2"/><i><font color="Green" title="E13">A12</font></i>: 
<font color="Maroon" title="c3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font>
 <b>and </b><br/><a NAME="E4:14_1_2"/><i><font color="Green" title="E14">A13</font></i>: 
<font color="Maroon" title="c2">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">e</font> where <font color="Olive" title="b1">e</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4" title="EQUATION:func.4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">s</font> : for <font color="Olive" title="b2">A</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">A</font>] holds <br/><font color="Olive" title="b2">A</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Olive" title="b1">e</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="birkhoff.html#E7:14_1"><i><font color="Green" title="E12">A11</font></i></a></i>;<br/>
<b>let </b><font color="Maroon" title="c5">q</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="E5:14_1_2"/>
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">j</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4" title="EQUATION:func.4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">s</font><b> such that </b><br/><a NAME="E7:14_1_2"/><i><font color="Green" title="E15">A14</font></i>: 
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z</font>
 <b>and </b><br/><a NAME="E8:14_1_2"/>
for <font color="Olive" title="b1">A</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] holds <br/><font color="Olive" title="b1">A</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Maroon" title="c6">z</font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E4:14_1_2"><i><font color="Green" title="E14">A13</font></i></a></i>;<br/>
<b>thus </b><a NAME="E9:14_1_2"/>
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="equation.html#K4" title="EQUATION:func.4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">j</font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E3:14_1_2"><i><font color="Green" title="E13">A12</font></i></a>, <a class="txt" href="birkhoff.html#E7:14_1_2"><i><font color="Green" title="E15">A14</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c3">E</font> = <font color="Maroon" title="c2">E</font> as   <a href="equation.html#NM1" title="EQUATION:NM.1">EqualSet</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<b>take </b>
<font color="Maroon" title="c3">E</font>
;<br/>

<b>let </b><font color="Maroon" title="c4">A</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:14_1_3"/><i><font color="Green" title="E13">A15</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">A</font>]
 ;<br/><b>thus </b><a NAME="E2:14_1_3"/>
<font color="Maroon" title="c4">A</font> <a href="equation.html#R2" title="EQUATION:pred.2">|=</a> <font color="Maroon" title="c3">E</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">s1</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>(); <i><font color="Red">:: according to </font></i><a class="ref" href="equation.html#D6" title="EQUATION:def.6">EQUATION:def 6</a><br/><b>let </b><font color="Maroon" title="c6">e</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4" title="EQUATION:func.4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font>;<br/>



<b>assume </b><a NAME="E1:14_1_3_1"/><i><font color="Green" title="E14">A16</font></i>: 
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font>
 ;<br/>

<a NAME="E2:14_1_3_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c3">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E7:14_1"><i><font color="Green" title="E12">A11</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4" title="EQUATION:func.4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s1</font><b> such that </b><br/><a NAME="E4:14_1_3_1"/><i><font color="Green" title="E15">A17</font></i>: 
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 <b>and </b><br/><a NAME="E5:14_1_3_1"/><i><font color="Green" title="E16">A18</font></i>: 
for <font color="Olive" title="b1">A</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] holds <br/><font color="Olive" title="b1">A</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Maroon" title="c7">x</font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E1:14_1_3_1"><i><font color="Green" title="E14">A16</font></i></a></i>;<br/>
<b>let </b><font color="Maroon" title="c8">h</font> be   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="equation.html#K3" title="EQUATION:func.3">TermAlg</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<font color="Maroon" title="c4">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="equation.html#D5" title="EQUATION:def.5">EQUATION:def 5</a><br/>

<b>assume </b><a NAME="E6:14_1_3_1"/><i><font color="Green" title="E17">A19</font></i>: 
<font color="Maroon" title="c8">h</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="equation.html#K3" title="EQUATION:func.3">TermAlg</a> <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c4">A</font>
 ;<br/>

<a NAME="E7:14_1_3_1"/>
<font color="Maroon" title="c4">A</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Maroon" title="c7">x</font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E1:14_1_3"><i><font color="Green" title="E13">A15</font></i></a>, <a class="txt" href="birkhoff.html#E5:14_1_3_1"><i><font color="Green" title="E16">A18</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:14_1_3_1"/>
<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="c5">s1</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="c6">e</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c5">s1</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="c6">e</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="birkhoff.html#E4:14_1_3_1"><i><font color="Green" title="E15">A17</font></i></a>, <a class="txt" href="birkhoff.html#E6:14_1_3_1"><i><font color="Green" title="E17">A19</font></i></a>, <a class="ref" href="equation.html#D5" title="EQUATION:def.5">EQUATION:def 5</a></i>;<br/>


</div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E11:14_1"/><i><font color="Green" title="E13">A20</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="equation.html#R2" title="EQUATION:pred.2">|=</a> <font color="Maroon" title="c3">E</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()] <b>means </b>$1 <a href="equation.html#R2" title="EQUATION:pred.2">|=</a> <font color="Maroon" title="c3">E</font>;<br/>
<a NAME="E12:14_1"/><i><font color="Green" title="E14">A21</font></i>: 
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">A</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">B</font>]
 
<b>by </b><i><a class="ref" href="equation.html#T36" title="EQUATION:th.36">EQUATION:36</a></i>;<br/>
<a NAME="E13:14_1"/><i><font color="Green" title="E15">A22</font></i>: 
for <font color="Olive" title="b1">A</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">B</font> being <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Olive" title="b1">A</font>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">A</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">B</font>]
 
<b>by </b><i><a class="ref" href="equation.html#T34" title="EQUATION:th.34">EQUATION:34</a></i>;<br/>
<a NAME="E14:14_1"/><i><font color="Green" title="E16">A23</font></i>: 
for <font color="Olive" title="b1">I</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">F</font> being   <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Olive" title="b1">I</font>,<font color="Maroon">F<sub>1</sub></font>()  st ( for <font color="Olive" title="b3">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">I</font> holds <br/> ex <font color="Olive" title="b4">A</font> being   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive" title="b4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b4">A</font>] ) ) holds <br/><font color="Maroon">S<sub>2</sub></font>[ <a href="pralg_2.html#K15" title="PRALG_2:func.15">product</a> <font color="Olive" title="b2">F</font>]
 
<b>by </b><i><a class="ref" href="equation.html#T40" title="EQUATION:th.40">EQUATION:40</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">XX</font> = 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> ;<br/>
<b>consider </b><font color="Maroon" title="c6">K</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon" title="c7">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> ,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c6">K</font><b> such that </b><br/><a NAME="E16:14_1"/><i><font color="Green" title="E17">A24</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">K</font>]
 <b>and </b><br/><a NAME="E17:14_1"/><i><font color="Green" title="E18">A25</font></i>: 
for <font color="Olive" title="b1">B</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">G</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> ,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Olive" title="b1">B</font>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">B</font>] holds <br/> ex <font color="Olive" title="b3">H</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c6">K</font>,<font color="Olive" title="b1">B</font> st <br/>( <font color="Olive" title="b3">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c6">K</font>,<font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b3">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c7">F</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">G</font> &amp; ( for <font color="Olive" title="b4">W</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c6">K</font>,<font color="Olive" title="b1">B</font>  st <font color="Olive" title="b4">W</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c6">K</font>,<font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b4">W</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c7">F</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">G</font> holds <br/><font color="Olive" title="b3">H</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b4">W</font> ) )
 <b>from </b><i><a class="ref" href="birkhoff.html#S2" target="_self" title="BIRKHOFF:sch.2">BIRKHOFF:sch 2</a>(<a class="txt" href="birkhoff.html#E12:14_1"><i><font color="Green" title="E14">A21</font></i></a>, <a class="txt" href="birkhoff.html#E13:14_1"><i><font color="Green" title="E15">A22</font></i></a>, <a class="txt" href="birkhoff.html#E14:14_1"><i><font color="Green" title="E16">A23</font></i></a>);<br/></i>
<a NAME="E18:14_1"/><i><font color="Green" title="E19">A26</font></i>: 
for <font color="Olive" title="b1">D</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">D</font>] iff for <font color="Olive" title="b2">s</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b3">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4" title="EQUATION:func.4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s</font>  st ( for <font color="Olive" title="b4">M</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b4">M</font>] holds <br/><font color="Olive" title="b4">M</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Olive" title="b3">e</font> ) holds <br/><font color="Olive" title="b1">D</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Olive" title="b3">e</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">D</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>thus </b><a NAME="E1:14_1_4"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">D</font>] implies for <font color="Olive" title="b1">s</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4" title="EQUATION:func.4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font>  st ( for <font color="Olive" title="b3">B</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b3">B</font>] holds <br/><font color="Olive" title="b3">B</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Olive" title="b2">e</font> ) holds <br/><font color="Maroon" title="c8">D</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Olive" title="b2">e</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1_4_1"/><i><font color="Green" title="E20">A27</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">D</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c9">s</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon" title="c10">e</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4" title="EQUATION:func.4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font>;<br/>



<b>assume </b><a NAME="E2:14_1_4_1"/><i><font color="Green" title="E21">A28</font></i>: 
for <font color="Olive" title="b1">B</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">B</font>] holds <br/><font color="Olive" title="b1">B</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Maroon" title="c10">e</font>
 ;<br/>

<a NAME="E3:14_1_4_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">s</font>,<font color="Maroon" title="c3">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E7:14_1"><i><font color="Green" title="E12">A11</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">s1</font> being   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E5:14_1_4_1"/><i><font color="Green" title="E22">A29</font></i>: 
( <font color="Maroon" title="c9">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s1</font> &amp; <font color="Maroon" title="c3">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4" title="EQUATION:func.4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font> : for <font color="Olive" title="b2">A</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">A</font>] holds <br/><font color="Olive" title="b2">A</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Olive" title="b1">f</font> </span>}</span>  )
 ;<br/>
<a NAME="E6:14_1_4_1"/>
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E2:14_1_4_1"><i><font color="Green" title="E21">A28</font></i></a>, <a class="txt" href="birkhoff.html#E5:14_1_4_1"><i><font color="Green" title="E22">A29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:14_1_4_1"/>
<font color="Maroon" title="c8">D</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Maroon" title="c10">e</font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E1:14_1_4_1"><i><font color="Green" title="E20">A27</font></i></a>, <a class="ref" href="equation.html#D6" title="EQUATION:def.6">EQUATION:def 6</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:14_1_4"/><i><font color="Green" title="E20">A30</font></i>: 
for <font color="Olive" title="b1">s</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4" title="EQUATION:func.4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font>  st ( for <font color="Olive" title="b3">B</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b3">B</font>] holds <br/><font color="Olive" title="b3">B</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Olive" title="b2">e</font> ) holds <br/><font color="Maroon" title="c8">D</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Olive" title="b2">e</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c9">s</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>(); <i><font color="Red">:: according to </font></i><a class="ref" href="equation.html#D6" title="EQUATION:def.6">EQUATION:def 6</a><br/><b>let </b><font color="Maroon" title="c10">e</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4" title="EQUATION:func.4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font>;<br/>



<b>assume </b><a NAME="E3:14_1_4"/><i><font color="Green" title="E21">A31</font></i>: 
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font>
 ;<br/>

<a NAME="E4:14_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">s</font>,<font color="Maroon" title="c3">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E7:14_1"><i><font color="Green" title="E12">A11</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">s1</font> being   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E6:14_1_4"/><i><font color="Green" title="E22">A32</font></i>: 
( <font color="Maroon" title="c9">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s1</font> &amp; <font color="Maroon" title="c3">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4" title="EQUATION:func.4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font> : for <font color="Olive" title="b2">A</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">A</font>] holds <br/><font color="Olive" title="b2">A</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Olive" title="b1">f</font> </span>}</span>  )
 ;<br/>
<b>consider </b><font color="Maroon" title="c12">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4" title="EQUATION:func.4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font><b> such that </b><br/><a NAME="E8:14_1_4"/><i><font color="Green" title="E23">A33</font></i>: 
( <font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> &amp; ( for <font color="Olive" title="b1">A</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] holds <br/><font color="Olive" title="b1">A</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Maroon" title="c12">f</font> ) )
 <b>by </b><i><a class="txt" href="birkhoff.html#E3:14_1_4"><i><font color="Green" title="E21">A31</font></i></a>, <a class="txt" href="birkhoff.html#E6:14_1_4"><i><font color="Green" title="E22">A32</font></i></a></i>;<br/>
<b>thus </b><a NAME="E9:14_1_4"/>
<font color="Maroon" title="c8">D</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Maroon" title="c10">e</font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E2:14_1_4"><i><font color="Green" title="E20">A30</font></i></a>, <a class="txt" href="birkhoff.html#E8:14_1_4"><i><font color="Green" title="E23">A33</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:14_1"/><i><font color="Green" title="E20">A34</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">K</font>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E16:14_1"><i><font color="Green" title="E17">A24</font></i></a></i>;<br/>
<a NAME="E20:14_1"/><i><font color="Green" title="E21">A35</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">K</font>]
 
<b>from </b><i><a class="ref" href="birkhoff.html#S11" target="_self" title="BIRKHOFF:sch.11">BIRKHOFF:sch 11</a>(<a class="txt" href="birkhoff.html#E18:14_1"><i><font color="Green" title="E19">A26</font></i></a>, <a class="txt" href="birkhoff.html#E17:14_1"><i><font color="Green" title="E18">A25</font></i></a>, <a class="txt" href="birkhoff.html#E19:14_1"><i><font color="Green" title="E20">A34</font></i></a>, <a class="txt" href="birkhoff.html#E1:14_1"><i><font color="Green" title="E7">A5</font></i></a>, <a class="txt" href="birkhoff.html#E2:14_1"><i><font color="Green" title="E8">A6</font></i></a>, <a class="txt" href="birkhoff.html#E4:14_1"><i><font color="Green" title="E10">A8</font></i></a>);<br/></i>
<a NAME="E21:14_1"/><i><font color="Green" title="E22">A36</font></i>: 
for <font color="Olive" title="b1">M</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">G</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> ,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Olive" title="b1">M</font>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">M</font>] holds <br/> ex <font color="Olive" title="b3">H</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c6">K</font>,<font color="Olive" title="b1">M</font> st <br/>( <font color="Olive" title="b3">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c6">K</font>,<font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b2">G</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b3">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c7">F</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">M</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon" title="c9">G</font> be    <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" title="c8">M</font>;<br/>



<b>assume </b><a NAME="E1:14_1_5"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">M</font>]
 ;<br/>

<a NAME="E2:14_1_5"/><b>then </b>
 ex <font color="Olive" title="b1">H</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c8">M</font> st <br/>( <font color="Olive" title="b1">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c8">M</font> &amp; <font color="Olive" title="b1">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c7">F</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c9">G</font> &amp; ( for <font color="Olive" title="b2">W</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c8">M</font>  st <font color="Olive" title="b2">W</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c8">M</font> &amp; <font color="Olive" title="b2">W</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c7">F</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c9">G</font> holds <br/><font color="Olive" title="b1">H</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">W</font> ) )
 
<b>by </b><i><a class="txt" href="birkhoff.html#E17:14_1"><i><font color="Green" title="E18">A25</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:14_1_5"/>
 ex <font color="Olive" title="b1">H</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c8">M</font> st <br/>( <font color="Olive" title="b1">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c8">M</font> &amp; <font color="Olive" title="b1">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c7">F</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c9">G</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E22:14_1"/><i><font color="Green" title="E23">A37</font></i>: 
for <font color="Olive" title="b1">B</font> being <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> <a href="msafree2.html#V3" title="MSAFREE2:attr.3">finitely-generated</a>  <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c4">A</font> holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">B</font> be  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> <a href="msafree2.html#V3" title="MSAFREE2:attr.3">finitely-generated</a>  <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c4">A</font>;<br/>

<a NAME="E1:14_1_6"/><i><font color="Green" title="E24">A38</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">B</font>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E11:14_1"><i><font color="Green" title="E13">A20</font></i></a>, <a class="ref" href="equation.html#T34" title="EQUATION:th.34">EQUATION:34</a></i>;<br/>
<b>thus </b><a NAME="E2:14_1_6"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c8">B</font>]
 <b>from </b><i><a class="ref" href="birkhoff.html#S7" target="_self" title="BIRKHOFF:sch.7">BIRKHOFF:sch 7</a>(<a class="txt" href="birkhoff.html#E1:14_1_6"><i><font color="Green" title="E24">A38</font></i></a>, <a class="txt" href="birkhoff.html#E20:14_1"><i><font color="Green" title="E21">A35</font></i></a>, <a class="txt" href="birkhoff.html#E21:14_1"><i><font color="Green" title="E22">A36</font></i></a>, <a class="txt" href="birkhoff.html#E1:14_1"><i><font color="Green" title="E7">A5</font></i></a>, <a class="txt" href="birkhoff.html#E3:14_1"><i><font color="Green" title="E9">A7</font></i></a>);</i><br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E23:14_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">A</font>]
 <b>from </b><i><a class="ref" href="birkhoff.html#S9" target="_self" title="BIRKHOFF:sch.9">BIRKHOFF:sch 9</a>(<a class="txt" href="birkhoff.html#E22:14_1"><i><font color="Green" title="E23">A37</font></i></a>, <a class="txt" href="birkhoff.html#E1:14_1"><i><font color="Green" title="E7">A5</font></i></a>, <a class="txt" href="birkhoff.html#E2:14_1"><i><font color="Green" title="E8">A6</font></i></a>, <a class="txt" href="birkhoff.html#E3:14_1"><i><font color="Green" title="E9">A7</font></i></a>, <a class="txt" href="birkhoff.html#E4:14_1"><i><font color="Green" title="E10">A8</font></i></a>);</i><br/>


</div>
