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

<a NAME="E1:11_1"/><span class="lab"><font color="Green" title="E8">A6</font></span>: 
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11"><span class="lab"><font color="Green" title="E4">A2</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c1">T</font> being   <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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">F<sub>2</sub></font>();<br/>
<span class="kw">set </span><font color="Maroon" title="c2">CC</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon">F<sub>2</sub></font>() :  ex <font color="Olive" title="b2">R</font> being  <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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">F<sub>2</sub></font>() st <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> </span>}</span> ;<br/>
<a NAME="E3:11_1"/>
<font color="Maroon" title="c1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon">F<sub>2</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D20" title="MSUALG_2:def.20">MSUALG_2:def 20</a></span>;<br/>
<a NAME="E4:11_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon">F<sub>2</sub></font>() :  ex <font color="Olive" title="b2">R</font> being  <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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">F<sub>2</sub></font>() st <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> </span>}</span> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">CC</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon">F<sub>2</sub></font>() :  ex <font color="Olive" title="b2">R</font> being  <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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">F<sub>2</sub></font>() st <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> </span>}</span>  as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<span class="kw">defpred </span><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> ] <span class="kw">means </span>$1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2;<br/>
<a NAME="E6:11_1"/><span class="lab"><font color="Green" title="E9">A7</font></span>: 
for <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">CC</font> holds <br/> ex <font color="Olive" title="b2">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">j</font>]
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">FF</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c3">CC</font><span class="kw"> such that </span><br/><a NAME="E8:11_1"/><span class="lab"><font color="Green" title="E10">A8</font></span>: 
for <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">CC</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">c</font>,<font color="Maroon" title="c4">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">c</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="pboole.html#S3" title="PBOOLE:sch.3">PBOOLE:sch 3</a>(<a class="txt" href="#E6:11_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>);<br/></span>
<a NAME="E9:11_1"/>
<font color="Maroon" title="c4">FF</font> is    <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c3">CC</font>,<font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="pralg_2.html#D12" title="PRALG_2:def.12">PRALG_2:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">CC</font> or <font color="Maroon" title="c4">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">c</font> is    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>() )</span><br/>

<span class="kw">assume </span><a NAME="E1:11_1_1"/><span class="lab"><font color="Green" title="E11">A9</font></span>: 
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">CC</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">c</font> is    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()</span><br/>

<a NAME="E2:11_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">Q</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Q</font> &amp;  ex <font color="Olive" title="b2">R</font> being  <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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">F<sub>2</sub></font>() st <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Q</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:11_1_1"/>
<font color="Maroon" title="c4">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">c</font> is    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:11_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="txt" href="#E1:11_1_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">FF</font> = <font color="Maroon" title="c4">FF</font> as    <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c3">CC</font>,<font color="Maroon">F<sub>1</sub></font>() ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">prSOR</font> being   <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of  <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c5">FF</font><span class="kw"> such that </span><br/><a NAME="E12:11_1"/><span class="lab"><font color="Green" title="E11">A10</font></span>: 
 ex <font color="Olive" title="b1">BA</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c6">prSOR</font>,<font color="Maroon">F<sub>2</sub></font>() st <font color="Olive" title="b1">BA</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon" title="c6">prSOR</font>,<font color="Maroon">F<sub>2</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:11_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="ref" href="equation.html#T29" title="EQUATION:th.29">EQUATION:29</a></span>;<br/>
<a NAME="E13:11_1"/><span class="lab"><font color="Green" title="E12">A11</font></span>: 
for <font color="Olive" title="b1">A</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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#K13" title="MSUALG_4:func.13">QuotMSAlg</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">R</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11"><span class="lab"><font color="Green" title="E6">A4</font></span></a></span>;<br/>
<a NAME="E14:11_1"/>
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> <font color="Maroon" title="c3">CC</font> holds <br/> ex <font color="Olive" title="b2">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="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">A</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">CC</font> implies  ex <font color="Olive" title="b1">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="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:11_1_2"/><span class="lab"><font color="Green" title="E13">A12</font></span>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">CC</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">Q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <font color="Maroon">F<sub>2</sub></font>()<span class="kw"> such that </span><br/><a NAME="E3:11_1_2"/><span class="lab"><font color="Green" title="E14">A13</font></span>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Q</font>
 <span class="kw">and </span><br/><a NAME="E4:11_1_2"/><span class="lab"><font color="Green" title="E15">A14</font></span>: 
 ex <font color="Olive" title="b1">R</font> being  <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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">F<sub>2</sub></font>() st <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Q</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">R</font> being   <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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">F<sub>2</sub></font>()<span class="kw"> such that </span><br/><a NAME="E6:11_1_2"/><span class="lab"><font color="Green" title="E16">A15</font></span>: 
<font color="Maroon" title="c9">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11_1_2"><span class="lab"><font color="Green" title="E15">A14</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">R</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c9">R</font>] )</span><br/>

<span class="kw">thus </span><a NAME="E7:11_1_2"/>
( <font color="Maroon" title="c9">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c9">R</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E8:11_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="txt" href="#E1:11_1_2"><span class="lab"><font color="Green" title="E13">A12</font></span></a>, <a class="txt" href="#E3:11_1_2"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="txt" href="#E6:11_1_2"><span class="lab"><font color="Green" title="E16">A15</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E15:11_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A16</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">prSOR</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E5:11"><span class="lab"><font color="Green" title="E7">A5</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E16:11_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>()]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="birkhoff.html#S8" target="_self" title="BIRKHOFF:sch.8">BIRKHOFF:sch 8</a>(<a class="txt" href="#E12:11_1"><span class="lab"><font color="Green" title="E11">A10</font></span></a>, <a class="txt" href="#E15:11_1"><span class="lab"><font color="Green" title="E13">A16</font></span></a>, <a class="txt" href="#E1:11_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E13:11_1"><span class="lab"><font color="Green" title="E12">A11</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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