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

<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon" title="c2">I</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">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" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c4">F</font> be    <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c1">S</font>;<br/>







<b>assume </b><a NAME="E1:32"/><i><font color="Green" title="E17">A1</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font>  ex <font color="Olive" title="b2">C</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="c3">A</font> st <font color="Olive" title="b2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b1">i</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c5">J</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>;<br/>
<b>set </b><font color="Maroon" title="c6">Z</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" title="c3">A</font> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> ex <font color="Olive" title="b3">D</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="c3">A</font> st <br/>( <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">D</font> ) </span>}</span> ;<br/>
<b>consider </b><font color="Maroon" title="c7">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font>;<br/>
<b>consider </b><font color="Maroon" title="c8">W</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="c3">A</font><b> such that </b><br/><a NAME="E4:32"/><i><font color="Green" title="E18">A2</font></i>: 
<font color="Maroon" title="c8">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c7">m</font>
 <b>by </b><i><a class="txt" href="equation.html#E1:32"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/>
<a NAME="E5:32"/>
<font color="Maroon" title="c8">W</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" title="c3">A</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D20" title="MSUALG_2:def.20">MSUALG_2:def 20</a></i>;<br/>
<a NAME="E6:32"/><b>then </b>
<font color="Maroon" title="c8">W</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" title="c3">A</font> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> ex <font color="Olive" title="b3">D</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="c3">A</font> st <br/>( <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">D</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="equation.html#E4:32"><i><font color="Green" title="E18">A2</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">Z</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" title="c3">A</font> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> ex <font color="Olive" title="b3">D</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="c3">A</font> st <br/>( <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">D</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/>
<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">Q</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="Maroon" title="c3">A</font> ex <font color="Olive" title="b2">G</font> being   <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Olive" title="b1">Q</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Q</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">G</font> &amp; <font color="Olive" title="b2">G</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> &amp; <font color="Olive" title="b2">G</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a> );<br/>
<a NAME="E8:32"/><i><font color="Green" title="E19">A3</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">Z</font>  ex <font color="Olive" title="b2">b</font> being   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">Z</font>;<br/>

<a NAME="E1:32_1"/>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c11">C</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" title="c3">A</font><b> such that </b><br/><a NAME="E3:32_1"/><i><font color="Green" title="E20">A4</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">C</font>
 <b>and </b><br/><a NAME="E4:32_1"/><i><font color="Green" title="E21">A5</font></i>: 
 ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> ex <font color="Olive" title="b2">D</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="c3">A</font> st <br/>( <font color="Maroon" title="c11">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c11">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">D</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c12">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font>, <font color="Maroon" title="c13">D</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="c3">A</font><b> such that </b><br/><a NAME="E6:32_1"/><i><font color="Green" title="E22">A6</font></i>: 
( <font color="Maroon" title="c11">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c11">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">D</font> )
 <b>by </b><i><a class="txt" href="equation.html#E4:32_1"><i><font color="Green" title="E21">A5</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">G</font> being    <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c13">D</font><b> such that </b><br/><a NAME="E8:32_1"/><i><font color="Green" title="E23">A7</font></i>: 
<font color="Maroon" title="c14">G</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a>
 <b>by </b><i><a class="ref" href="msafree2.html#D10" title="MSAFREE2:def.10">MSAFREE2:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">S</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a> <a href="finset_1.html#V2" title="FINSET_1:attr.2">V19</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c13">D</font><b> such that </b><br/><a NAME="E10:32_1"/><i><font color="Green" title="E24">A8</font></i>: 
<font color="Maroon" title="c14">G</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c15">S</font>
 <b>by </b><i><a class="txt" href="equation.html#E8:32_1"><i><font color="Green" title="E23">A7</font></i></a>, <a class="ref" href="msualg_9.html#T8" title="MSUALG_9:th.8">MSUALG_9:8</a></i>;<br/>
<a NAME="E11:32_1"/>
<font color="Maroon" title="c15">S</font> is    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_1_1"/><i><font color="Green" title="E25">A9</font></i>: 
<font color="Maroon" title="c15">S</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c13">D</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E2:32_1_1"/>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c13">D</font> is   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D10" title="MSUALG_2:def.10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E3:32_1_1"/><b>then </b>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c13">D</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<b>hence </b><a NAME="E4:32_1_1"/>
<font color="Maroon" title="c15">S</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="equation.html#E1:32_1_1"><i><font color="Green" title="E25">A9</font></i></a>, <a class="ref" href="pboole.html#T15" title="PBOOLE:th.15">PBOOLE:15</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c16">b</font> = <font color="Maroon" title="c15">S</font> as    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <b>by </b><i><a class="ref" href="closure2.html#D1" title="CLOSURE2:def.1">CLOSURE2:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c16">b</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c13">D</font>
;<br/>

<b>reconsider </b><font color="Maroon" title="c17">S</font> = <font color="Maroon" title="c15">S</font> as    <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c13">D</font> <b>by </b><i><a class="txt" href="equation.html#E10:32_1"><i><font color="Green" title="E24">A8</font></i></a>, <a class="ref" href="msscyc_1.html#T21" title="MSSCYC_1:th.21">MSSCYC_1:21</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c17">S</font>
;<br/>

<b>thus </b><a NAME="E14:32_1"/>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">D</font>
 <b>by </b><i><a class="txt" href="equation.html#E3:32_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="equation.html#E6:32_1"><i><font color="Green" title="E22">A6</font></i></a></i>;<br/>

<b>thus </b><a NAME="E15:32_1"/>
( <font color="Maroon" title="c16">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">S</font> &amp; <font color="Maroon" title="c17">S</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> &amp; <font color="Maroon" title="c17">S</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c10">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c9">Z</font>, <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font><b> such that </b><br/><a NAME="E10:32"/><i><font color="Green" title="E20">A10</font></i>: 
for <font color="Olive" title="b1">B</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">Z</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>,<font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">B</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="equation.html#E8:32"><i><font color="Green" title="E19">A3</font></i></a>);<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> ex <font color="Olive" title="b3">K</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" title="c1">S</font> st <br/>( <font color="Olive" title="b3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b2">i</font></span>)</span> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 ) </span>}</span> ;<br/>
<b>consider </b><font color="Maroon" title="c11">SOR</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" title="c1">S</font><b> such that </b><br/><a NAME="E12:32"/><i><font color="Green" title="E21">A11</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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" title="c1">S</font> holds <br/><font color="Maroon" title="c11">SOR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">j</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S4" title="PBOOLE:sch.4">PBOOLE:sch 4</a>();<br/></i>
<a NAME="E13:32"/>
<font color="Maroon" title="c11">SOR</font> is    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">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="E1:32_2"/><i><font color="Green" title="E22">A12</font></i>: 
<font color="Maroon" title="c12">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" title="c1">S</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c13">UU</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> ex <font color="Olive" title="b3">K</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" title="c1">S</font> st <br/>( <font color="Olive" title="b3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b2">i</font></span>)</span> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> ) </span>}</span> ;<br/>
<b>let </b><font color="Maroon" title="c14">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="E2:32_2"/>
<font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">SOR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font>
 ;<br/>

<a NAME="E3:32_2"/><b>then </b>
<font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> ex <font color="Olive" title="b3">K</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" title="c1">S</font> st <br/>( <font color="Olive" title="b3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b2">i</font></span>)</span> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="equation.html#E12:32"><i><font color="Green" title="E21">A11</font></i></a>, <a class="txt" href="equation.html#E1:32_2"><i><font color="Green" title="E22">A12</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:32_2"/><i><font color="Green" title="E23">A13</font></i>: 
<font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">w</font>
 <b>and </b><br/><a NAME="E6:32_2"/><i><font color="Green" title="E24">A14</font></i>: 
<font color="Maroon" title="c15">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> ex <font color="Olive" title="b3">K</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" title="c1">S</font> st <br/>( <font color="Olive" title="b3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b2">i</font></span>)</span> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span><b> such that </b><br/><a NAME="E8:32_2"/><i><font color="Green" title="E25">A15</font></i>: 
<font color="Maroon" title="c15">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a</font>
 <b>and </b><br/><a NAME="E9:32_2"/>
 ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> ex <font color="Olive" title="b2">K</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" title="c1">S</font> st <br/>( <font color="Olive" title="b2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b1">i</font></span>)</span> &amp; <font color="Maroon" title="c16">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> )
 <b>by </b><i><a class="txt" href="equation.html#E6:32_2"><i><font color="Green" title="E24">A14</font></i></a></i>;<br/>
<b>thus </b><a NAME="E10:32_2"/>
<font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font>
 <b>by </b><i><a class="txt" href="equation.html#E5:32_2"><i><font color="Green" title="E23">A13</font></i></a>, <a class="txt" href="equation.html#E8:32_2"><i><font color="Green" title="E25">A15</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c12">SOR</font> = <font color="Maroon" title="c11">SOR</font> as   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c3">A</font> ;<br/>
<div><i><font color="Green" title="E22">A16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:32_3"/><i><font color="Green" title="E23">A17</font></i>: 
<font color="Maroon" title="c13">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" title="c1">S</font>
 ;<br/><b>set </b><font color="Maroon" title="c14">UU</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> ex <font color="Olive" title="b3">K</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" title="c1">S</font> st <br/>( <font color="Olive" title="b3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b2">i</font></span>)</span> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> ) </span>}</span> ;<br/><b>consider </b><font color="Maroon" title="c15">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font>;<br/><b>consider </b><font color="Maroon" title="c16">C</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="c3">A</font><b> such that </b><br/><a NAME="E4:32_3"/><i><font color="Green" title="E24">A18</font></i>: 
<font color="Maroon" title="c16">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c15">i</font>
 <b>by </b><i><a class="txt" href="equation.html#E1:32"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/><a NAME="E5:32_3"/>
<font color="Maroon" title="c16">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" title="c3">A</font>
 <b>by </b><i><a class="ref" href="msualg_2.html#D20" title="MSUALG_2:def.20">MSUALG_2:def 20</a></i>;<br/><a NAME="E6:32_3"/><b>then </b><i><font color="Green" title="E25">A19</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font>
 <b>by </b><i><a class="txt" href="equation.html#E4:32_3"><i><font color="Green" title="E24">A18</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">Q</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="Maroon" title="c3">A</font><b> such that </b><br/><a NAME="E8:32_3"/><i><font color="Green" title="E26">A20</font></i>: 
 ex <font color="Olive" title="b1">G</font> being   <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c17">Q</font> st <br/>( <font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Q</font> &amp; <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c15">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G</font> &amp; <font color="Olive" title="b1">G</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> &amp; <font color="Olive" title="b1">G</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a> )
 <b>by </b><i><a class="txt" href="equation.html#E10:32"><i><font color="Green" title="E20">A10</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">G1</font> = <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c15">i</font></span>)</span> as  <a href="finset_1.html#V2" title="FINSET_1:attr.2">V19</a>  <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <b>by </b><i><a class="txt" href="equation.html#E6:32_3"><i><font color="Green" title="E25">A19</font></i></a>, <a class="txt" href="equation.html#E8:32_3"><i><font color="Green" title="E26">A20</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/><a NAME="E10:32_3"/>
<font color="Maroon" title="c18">G1</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/><a NAME="E11:32_3"/><b>then </b>
<font color="Maroon" title="c18">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font>
 <b>by </b><i><a class="txt" href="equation.html#E1:32_3"><i><font color="Green" title="E23">A17</font></i></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/><a NAME="E12:32_3"/><b>then </b>
<font color="Maroon" title="c18">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> ex <font color="Olive" title="b3">K</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" title="c1">S</font> st <br/>( <font color="Olive" title="b3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b2">i</font></span>)</span> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> ) </span>}</span> 
 ;<br/><b>hence </b><a NAME="E13:32_3"/>
not <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> ex <font color="Olive" title="b3">K</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" title="c1">S</font> st <br/>( <font color="Olive" title="b3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b2">i</font></span>)</span> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> ) </span>}</span>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/></div><b>end;</b></div>
<a NAME="E16:32"/>
<font color="Maroon" title="c12">SOR</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">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#D16" title="PBOOLE:def.16">PBOOLE:def 16</a><br/>

<b>assume </b><a NAME="E1:32_4"/><i><font color="Green" title="E23">A21</font></i>: 
<font color="Maroon" title="c13">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" title="c1">S</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c14">UU</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> ex <font color="Olive" title="b3">K</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" title="c1">S</font> st <br/>( <font color="Olive" title="b3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b2">i</font></span>)</span> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> ) </span>}</span> ;<br/>
<b>consider </b><font color="Maroon" title="c15">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font>;<br/>
<b>consider </b><font color="Maroon" title="c16">C</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="c3">A</font><b> such that </b><br/><a NAME="E4:32_4"/><i><font color="Green" title="E24">A22</font></i>: 
<font color="Maroon" title="c16">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c15">i</font>
 <b>by </b><i><a class="txt" href="equation.html#E1:32"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/>
<a NAME="E5:32_4"/>
<font color="Maroon" title="c16">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" title="c3">A</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D20" title="MSUALG_2:def.20">MSUALG_2:def 20</a></i>;<br/>
<a NAME="E6:32_4"/><b>then </b><i><font color="Green" title="E25">A23</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font>
 
<b>by </b><i><a class="txt" href="equation.html#E4:32_4"><i><font color="Green" title="E24">A22</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">Q</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="Maroon" title="c3">A</font><b> such that </b><br/><a NAME="E8:32_4"/><i><font color="Green" title="E26">A24</font></i>: 
 ex <font color="Olive" title="b1">G</font> being   <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c17">Q</font> st <br/>( <font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Q</font> &amp; <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c15">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G</font> &amp; <font color="Olive" title="b1">G</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> &amp; <font color="Olive" title="b1">G</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a> )
 <b>by </b><i><a class="txt" href="equation.html#E10:32"><i><font color="Green" title="E20">A10</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">G1</font> = <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c15">i</font></span>)</span> as  <a href="finset_1.html#V2" title="FINSET_1:attr.2">V19</a>  <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <b>by </b><i><a class="txt" href="equation.html#E6:32_4"><i><font color="Green" title="E25">A23</font></i></a>, <a class="txt" href="equation.html#E8:32_4"><i><font color="Green" title="E26">A24</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<a NAME="E10:32_4"/>
<font color="Maroon" title="c18">G1</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E11:32_4"/><b>then </b>
<font color="Maroon" title="c18">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font>
 
<b>by </b><i><a class="txt" href="equation.html#E1:32_4"><i><font color="Green" title="E23">A21</font></i></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E12:32_4"/><b>then </b><i><font color="Green" title="E27">A25</font></i>: 
<font color="Maroon" title="c18">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> ex <font color="Olive" title="b3">K</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" title="c1">S</font> st <br/>( <font color="Olive" title="b3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b2">i</font></span>)</span> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> ) </span>}</span> 
 
;<br/>
<a NAME="E13:32_4"/>
not <font color="Maroon" title="c18">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="equation.html#E1:32_4"><i><font color="Green" title="E23">A21</font></i></a>, <a class="txt" href="equation.html#E8:32_4"><i><font color="Green" title="E26">A24</font></i></a>, <a class="ref" href="pboole.html#D16" title="PBOOLE:def.16">PBOOLE:def 16</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c19">q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E15:32_4"/><i><font color="Green" title="E28">A26</font></i>: 
<font color="Maroon" title="c19">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E16:32_4"/>
<font color="Maroon" title="c19">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font></span>)</span> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> ex <font color="Olive" title="b3">K</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" title="c1">S</font> st <br/>( <font color="Olive" title="b3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b2">i</font></span>)</span> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="equation.html#E12:32_4"><i><font color="Green" title="E27">A25</font></i></a>, <a class="txt" href="equation.html#E15:32_4"><i><font color="Green" title="E28">A26</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E17:32_4"/>
not <font color="Maroon" title="c12">SOR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="equation.html#E12:32"><i><font color="Green" title="E21">A11</font></i></a>, <a class="txt" href="equation.html#E1:32_4"><i><font color="Green" title="E23">A21</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c13">SOR</font> = <font color="Maroon" title="c12">SOR</font> as  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a> <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c3">A</font> ;<br/>
<a NAME="E18:32"/>
<font color="Maroon" title="c13">SOR</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">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="pre_circ.html#D3" title="PRE_CIRC:def.3">PRE_CIRC:def 3</a><br/>

<b>assume </b><a NAME="E1:32_5"/><i><font color="Green" title="E23">A27</font></i>: 
<font color="Maroon" title="c14">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" title="c1">S</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c15">UU</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> ex <font color="Olive" title="b3">K</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" title="c1">S</font> st <br/>( <font color="Olive" title="b3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b2">i</font></span>)</span> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> ) </span>}</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c16">VV</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> ex <font color="Olive" title="b3">K</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" title="c1">S</font> st <br/>( <font color="Olive" title="b3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b2">i</font></span>)</span> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</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>  <b>by </b><i><a class="txt" href="equation.html#E15:32"><i><font color="Green" title="E22">A16</font></i></a>, <a class="txt" href="equation.html#E1:32_5"><i><font color="Green" title="E23">A27</font></i></a></i>;<br/>
<div><i><font color="Green" title="E24">A28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_5_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c17">r</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E2:32_5_1"/><i><font color="Green" title="E25">A29</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">I</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T73" title="FINSEQ_1:th.73">FINSEQ_1:73</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</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">L</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" title="c1">S</font> st <br/>( <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">L</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> );<br/><a NAME="E3:32_5_1"/><i><font color="Green" title="E26">A30</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font>  ex <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">VV</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_5_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font>;<br/>

<b>consider </b><font color="Maroon" title="c19">W</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="c3">A</font><b> such that </b><br/><a NAME="E2:32_5_1_1"/><i><font color="Green" title="E27">A31</font></i>: 
<font color="Maroon" title="c19">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c18">a</font>
 <b>by </b><i><a class="txt" href="equation.html#E1:32"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/>
<a NAME="E3:32_5_1_1"/>
<font color="Maroon" title="c19">W</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" title="c3">A</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D20" title="MSUALG_2:def.20">MSUALG_2:def 20</a></i>;<br/>
<a NAME="E4:32_5_1_1"/><b>then </b><i><font color="Green" title="E28">A32</font></i>: 
<font color="Maroon" title="c19">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font>
 
<b>by </b><i><a class="txt" href="equation.html#E2:32_5_1_1"><i><font color="Green" title="E27">A31</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c20">Q</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="Maroon" title="c3">A</font><b> such that </b><br/><a NAME="E6:32_5_1_1"/><i><font color="Green" title="E29">A33</font></i>: 
 ex <font color="Olive" title="b1">G</font> being   <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c20">Q</font> st <br/>( <font color="Maroon" title="c19">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Q</font> &amp; <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G</font> &amp; <font color="Olive" title="b1">G</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> &amp; <font color="Olive" title="b1">G</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a> )
 <b>by </b><i><a class="txt" href="equation.html#E10:32"><i><font color="Green" title="E20">A10</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">G1</font> = <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c18">a</font></span>)</span> as  <a href="finset_1.html#V2" title="FINSET_1:attr.2">V19</a>  <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <b>by </b><i><a class="txt" href="equation.html#E2:32_5_1_1"><i><font color="Green" title="E27">A31</font></i></a>, <a class="txt" href="equation.html#E4:32_5_1_1"><i><font color="Green" title="E28">A32</font></i></a>, <a class="txt" href="equation.html#E6:32_5_1_1"><i><font color="Green" title="E29">A33</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<a NAME="E8:32_5_1_1"/>
<font color="Maroon" title="c21">G1</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E9:32_5_1_1"/><b>then </b>
<font color="Maroon" title="c21">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font>
 
<b>by </b><i><a class="txt" href="equation.html#E1:32_5"><i><font color="Green" title="E23">A27</font></i></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E10:32_5_1_1"/><b>then </b>
<font color="Maroon" title="c21">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> ex <font color="Olive" title="b3">K</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" title="c1">S</font> st <br/>( <font color="Olive" title="b3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b2">i</font></span>)</span> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> ) </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c22">b</font> = <font color="Maroon" title="c21">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">VV</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c22">b</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c21">G1</font>
;<br/>

<b>thus </b><a NAME="E12:32_5_1_1"/>
( <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">G1</font> &amp; <font color="Maroon" title="c22">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> )
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c18">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c16">VV</font><b> such that </b><br/><a NAME="E5:32_5_1"/><i><font color="Green" title="E27">A34</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c18">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="equation.html#E3:32_5_1"><i><font color="Green" title="E26">A30</font></i></a>);<br/></i><a NAME="E6:32_5_1"/><i><font color="Green" title="E28">A35</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">h</font>
 <b>by </b><i><a class="txt" href="equation.html#E2:32_5_1"><i><font color="Green" title="E25">A29</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c19">p</font> = <font color="Maroon" title="c18">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c17">r</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#T20" title="FINSEQ_1:th.20">FINSEQ_1:20</a></i>;<br/><b>take </b><font color="Maroon" title="c20">p</font> = <font color="Maroon" title="c19">p</font>;<br/><b>thus </b><a NAME="E8:32_5_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">VV</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_5_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:32_5_1_2"/><i><font color="Green" title="E29">A36</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">h</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_5_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">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="E1:32_5_1_2_1"/>
<font color="Maroon" title="c21">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">p</font>
 ;<br/>

<b>hence </b><a NAME="E2:32_5_1_2_1"/>
<font color="Maroon" title="c21">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">h</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T25" title="FUNCT_1:th.25">FUNCT_1:25</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E2:32_5_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">VV</font>
 <b>by </b><i><a class="txt" href="equation.html#E1:32_5_1_2"><i><font color="Green" title="E29">A36</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E3:32_5_1_2"/>
<font color="Maroon" title="c16">VV</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_5_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">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="E1:32_5_1_2_2"/>
<font color="Maroon" title="c21">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">VV</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c22">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span><b> such that </b><br/><a NAME="E3:32_5_1_2_2"/><i><font color="Green" title="E30">A37</font></i>: 
<font color="Maroon" title="c21">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">a</font>
 <b>and </b><br/><a NAME="E4:32_5_1_2_2"/><i><font color="Green" title="E31">A38</font></i>: 
 ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> ex <font color="Olive" title="b2">K</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" title="c1">S</font> st <br/>( <font color="Olive" title="b2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b1">i</font></span>)</span> &amp; <font color="Maroon" title="c22">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c23">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font>, <font color="Maroon" title="c24">K</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" title="c1">S</font><b> such that </b><br/><a NAME="E6:32_5_1_2_2"/><i><font color="Green" title="E32">A39</font></i>: 
( <font color="Maroon" title="c24">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c23">i</font></span>)</span> &amp; <font color="Maroon" title="c22">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> )
 <b>by </b><i><a class="txt" href="equation.html#E4:32_5_1_2_2"><i><font color="Green" title="E31">A38</font></i></a></i>;<br/>
<a NAME="E7:32_5_1_2_2"/><i><font color="Green" title="E33">A40</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">h</font>
 
<b>by </b><i><a class="txt" href="equation.html#E6:32_5_1"><i><font color="Green" title="E28">A35</font></i></a>, <a class="ref" href="relat_1.html#T47" title="RELAT_1:th.47">RELAT_1:47</a></i>;<br/>
<a NAME="E8:32_5_1_2_2"/><i><font color="Green" title="E34">A41</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">I</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:32_5_1_2_2"/>
 ex <font color="Olive" title="b1">L</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" title="c1">S</font> st <br/>( <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c23">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">L</font> &amp; <font color="Maroon" title="c18">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> )
 
<b>by </b><i><a class="txt" href="equation.html#E5:32_5_1"><i><font color="Green" title="E27">A34</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:32_5_1_2_2"/>
<font color="Maroon" title="c21">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">p</font>
 <b>by </b><i><a class="txt" href="equation.html#E3:32_5_1_2_2"><i><font color="Green" title="E30">A37</font></i></a>, <a class="txt" href="equation.html#E6:32_5_1_2_2"><i><font color="Green" title="E32">A39</font></i></a>, <a class="txt" href="equation.html#E7:32_5_1_2_2"><i><font color="Green" title="E33">A40</font></i></a>, <a class="txt" href="equation.html#E8:32_5_1_2_2"><i><font color="Green" title="E34">A41</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


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


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E4:32_5"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">a</font> where <font color="Olive" title="b2">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span> :  ex <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> ex <font color="Olive" title="b4">K</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" title="c1">S</font> st <br/>( <font color="Olive" title="b4">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b3">i</font></span>)</span> &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> ) </span>}</span>  holds <br/><font color="Olive" title="b1">x</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:32_5_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> ex <font color="Olive" title="b3">K</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" title="c1">S</font> st <br/>( <font color="Olive" title="b3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b2">i</font></span>)</span> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c18">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span><b> such that </b><br/><a NAME="E3:32_5_2"/><i><font color="Green" title="E25">A42</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">a</font>
 <b>and </b><br/><a NAME="E4:32_5_2"/><i><font color="Green" title="E26">A43</font></i>: 
 ex <font color="Olive" title="b1">w</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> ex <font color="Olive" title="b2">K</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" title="c1">S</font> st <br/>( <font color="Olive" title="b2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b1">w</font></span>)</span> &amp; <font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c19">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font>, <font color="Maroon" title="c20">K</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" title="c1">S</font><b> such that </b><br/><a NAME="E6:32_5_2"/><i><font color="Green" title="E27">A44</font></i>: 
( <font color="Maroon" title="c20">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c19">w</font></span>)</span> &amp; <font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> )
 <b>by </b><i><a class="txt" href="equation.html#E4:32_5_2"><i><font color="Green" title="E26">A43</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">E</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="c3">A</font><b> such that </b><br/><a NAME="E8:32_5_2"/><i><font color="Green" title="E28">A45</font></i>: 
<font color="Maroon" title="c21">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c19">w</font>
 <b>by </b><i><a class="txt" href="equation.html#E1:32"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/>
<a NAME="E9:32_5_2"/>
<font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c19">w</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" title="c3">A</font>
 
<b>by </b><i><a class="txt" href="equation.html#E8:32_5_2"><i><font color="Green" title="E28">A45</font></i></a>, <a class="ref" href="msualg_2.html#D20" title="MSUALG_2:def.20">MSUALG_2:def 20</a></i>;<br/>
<a NAME="E10:32_5_2"/><b>then </b>
<font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c19">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font>
 
<b>by </b><i><a class="txt" href="equation.html#E8:32_5_2"><i><font color="Green" title="E28">A45</font></i></a></i>;<br/>
<a NAME="E11:32_5_2"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c19">w</font>,<font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c19">w</font></span>)</span>]
 
<b>by </b><i><a class="txt" href="equation.html#E10:32"><i><font color="Green" title="E20">A10</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c22">Q</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="Maroon" title="c3">A</font><b> such that </b><br/><a NAME="E13:32_5_2"/>
<font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c19">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">Q</font>
 <b>and </b><br/><a NAME="E14:32_5_2"/><i><font color="Green" title="E29">A46</font></i>: 
 ex <font color="Olive" title="b1">G</font> being   <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c22">Q</font> st <br/>( <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c19">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G</font> &amp; <font color="Olive" title="b1">G</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a> )
 ;<br/>
<b>thus </b><a NAME="E15:32_5_2"/>
<font color="Maroon" title="c17">x</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="equation.html#E1:32_5"><i><font color="Green" title="E23">A27</font></i></a>, <a class="txt" href="equation.html#E3:32_5_2"><i><font color="Green" title="E25">A42</font></i></a>, <a class="txt" href="equation.html#E6:32_5_2"><i><font color="Green" title="E27">A44</font></i></a>, <a class="txt" href="equation.html#E14:32_5_2"><i><font color="Green" title="E29">A46</font></i></a>, <a class="ref" href="pre_circ.html#D3" title="PRE_CIRC:def.3">PRE_CIRC:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:32_5"/><b>then </b>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font></span>)</span> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> ex <font color="Olive" title="b3">K</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" title="c1">S</font> st <br/>( <font color="Olive" title="b3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b2">i</font></span>)</span> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> ) </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="equation.html#E3:32_5"><i><font color="Green" title="E24">A28</font></i></a>, <a class="ref" href="finset_1.html#T25" title="FINSET_1:th.25">FINSET_1:25</a></i>;<br/>
<b>hence </b><a NAME="E6:32_5"/>
<font color="Maroon" title="c13">SOR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="equation.html#E12:32"><i><font color="Green" title="E21">A11</font></i></a>, <a class="txt" href="equation.html#E1:32_5"><i><font color="Green" title="E23">A27</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c14">SOR</font> = <font color="Maroon" title="c13">SOR</font> as  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a> <a href="finset_1.html#V2" title="FINSET_1:attr.2">V19</a> <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c3">A</font> ;<br/>
<b>take </b><font color="Maroon" title="c15">B</font> =  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c14">SOR</font>;<br/>

<b>let </b><font color="Maroon" title="c16">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font>;<br/>

<b>consider </b><font color="Maroon" title="c17">C</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="c3">A</font><b> such that </b><br/><a NAME="E21:32"/><i><font color="Green" title="E23">A47</font></i>: 
<font color="Maroon" title="c17">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c16">i</font>
 <b>by </b><i><a class="txt" href="equation.html#E1:32"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/>
<a NAME="E22:32"/>
<font color="Maroon" title="c17">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" title="c3">A</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D20" title="MSUALG_2:def.20">MSUALG_2:def 20</a></i>;<br/>
<a NAME="E23:32"/><b>then </b>
<font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font>
 
<b>by </b><i><a class="txt" href="equation.html#E21:32"><i><font color="Green" title="E23">A47</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">Q</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="Maroon" title="c3">A</font>, <font color="Maroon" title="c19">G</font> being    <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c18">Q</font><b> such that </b><br/><a NAME="E25:32"/><i><font color="Green" title="E24">A48</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">Q</font>
 <b>and </b><br/><a NAME="E26:32"/><i><font color="Green" title="E25">A49</font></i>: 
( <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">G</font> &amp; <font color="Maroon" title="c19">G</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> &amp; <font color="Maroon" title="c19">G</font> is <a href="finset_1.html#V2" title="FINSET_1:attr.2">locally-finite</a> )
 <b>by </b><i><a class="txt" href="equation.html#E10:32"><i><font color="Green" title="E20">A10</font></i></a></i>;<br/>
<a NAME="E27:32"/><i><font color="Green" title="E26">A50</font></i>: 
<font color="Maroon" title="c19">G</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c18">Q</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E28:32"/>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c18">Q</font> is   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D10" title="MSUALG_2:def.10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E29:32"/><b>then </b>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c18">Q</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E30:32"/><b>then </b><i><font color="Green" title="E27">A51</font></i>: 
<font color="Maroon" title="c19">G</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="txt" href="equation.html#E27:32"><i><font color="Green" title="E26">A50</font></i></a>, <a class="ref" href="pboole.html#T15" title="PBOOLE:th.15">PBOOLE:15</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c20">G1</font> = <font color="Maroon" title="c19">G</font> as  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a> <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c3">A</font> <b>by </b><i><a class="txt" href="equation.html#E26:32"><i><font color="Green" title="E25">A49</font></i></a>, <a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E32:32"/><i><font color="Green" title="E28">A52</font></i>: 
<font color="Maroon" title="c20">G1</font> is    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of <font color="Maroon" title="c14">SOR</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">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="E1:32_6"/><i><font color="Green" title="E29">A53</font></i>: 
<font color="Maroon" title="c21">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" title="c1">S</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c22">UU</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j</font></span>)</span> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> ex <font color="Olive" title="b3">K</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" title="c1">S</font> st <br/>( <font color="Olive" title="b3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b2">i</font></span>)</span> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j</font> ) </span>}</span> ;<br/>
<b>let </b><font color="Maroon" title="c23">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="E2:32_6"/><i><font color="Green" title="E30">A54</font></i>: 
<font color="Maroon" title="c23">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j</font>
 ;<br/>

<a NAME="E3:32_6"/>
<font color="Maroon" title="c20">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j</font>
 
<b>by </b><i><a class="txt" href="equation.html#E30:32"><i><font color="Green" title="E27">A51</font></i></a>, <a class="txt" href="equation.html#E1:32_6"><i><font color="Green" title="E29">A53</font></i></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E4:32_6"/><b>then </b>
<font color="Maroon" title="c20">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j</font></span>)</span> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> ex <font color="Olive" title="b3">K</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" title="c1">S</font> st <br/>( <font color="Olive" title="b3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b2">i</font></span>)</span> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="equation.html#E26:32"><i><font color="Green" title="E25">A49</font></i></a></i>;<br/>
<a NAME="E5:32_6"/><b>then </b>
<font color="Maroon" title="c23">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j</font></span>)</span> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> ex <font color="Olive" title="b3">K</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" title="c1">S</font> st <br/>( <font color="Olive" title="b3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b2">i</font></span>)</span> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="equation.html#E2:32_6"><i><font color="Green" title="E30">A54</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:32_6"/>
<font color="Maroon" title="c23">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">SOR</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j</font>
 <b>by </b><i><a class="txt" href="equation.html#E12:32"><i><font color="Green" title="E21">A11</font></i></a>, <a class="txt" href="equation.html#E1:32_6"><i><font color="Green" title="E29">A53</font></i></a></i>;<br/>


</div><b>end;</b></div>
<font color="Maroon" title="c17">C</font> = 
 <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c19">G</font>
<b>by </b><i><a class="txt" href="equation.html#E21:32"><i><font color="Green" title="E23">A47</font></i></a>, <a class="txt" href="equation.html#E25:32"><i><font color="Green" title="E24">A48</font></i></a>, <a class="ref" href="msafree.html#T3" title="MSAFREE:th.3">MSAFREE:3</a></i>
<br/>.= 
 <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c20">G1</font>
<b>by </b><i><a class="ref" href="extens_1.html#T22" title="EXTENS_1:th.22">EXTENS_1:22</a></i>
;<br/>
<b>hence </b><a NAME="E34:32"/>
<font color="Maroon" title="c4">F</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c16">i</font> is    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c15">B</font>
 <b>by </b><i><a class="txt" href="equation.html#E21:32"><i><font color="Green" title="E23">A47</font></i></a>, <a class="txt" href="equation.html#E32:32"><i><font color="Green" title="E28">A52</font></i></a>, <a class="ref" href="extens_1.html#T21" title="EXTENS_1:th.21">EXTENS_1:21</a></i>;<br/>


</div>
