<?xml version="1.0"?>
<div><div><a NAME="S9"><span class="kw">scheme  </span><span class="comment"><font color="firebrick">:: BIRKHOFF:sch 9</font></span><br/></a><span class="lab"><font color="Green" title="E9">AllFinGen</font></span>{ <font color="Maroon">F<sub>1</sub></font>() <span class="kw">-&gt; </span>  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   non  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> , <font color="Maroon">F<sub>2</sub></font>() <span class="kw">-&gt; </span>  <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> over <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">P<sub>1</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] } :<br/><div class="add"><a NAME="E6:11"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>()]
 </div><span class="kw">provided</span><div class="add"><a NAME="E1:11"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
 for <font color="Olive" title="b1">B</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>() holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">B</font>]
 <span class="kw">and </span><br/><a NAME="E2:11"/><span class="lab"><font color="Green" title="E4">A2</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> over <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">and </span><br/><a NAME="E3:11"/><span class="lab"><font color="Green" title="E5">A3</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> over <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b2">B</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 <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>]
 <span class="kw">and </span><br/><a NAME="E4:11"/><span class="lab"><font color="Green" title="E6">A4</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> over <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b2">R</font> being   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">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">and </span><br/><a NAME="E5:11"/><span class="lab"><font color="Green" title="E7">A5</font></span>: 
 for <font color="Olive" title="b1">I</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">F</font> being    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">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#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b3">i</font> <a href="tarski.html#R2" title="TARSKI: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> over <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#K14" title="PRALG_2:func.14">product</a> <font color="Olive" title="b2">F</font>]
 </div></div></div>
