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

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>( $1 is    <a href="dynkin.html#M1" title="DYNKIN:mode.1">Dynkin_System</a> of <font color="Maroon" title="c1">Omega</font> &amp; <font color="Maroon" title="c2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> $1 );<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:30_1_1"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">Omega</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></span>
<a NAME="E3:30_1_1"/>
 <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">Omega</font> is    <a href="dynkin.html#M1" title="DYNKIN:mode.1">Dynkin_System</a> of <font color="Maroon" title="c1">Omega</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dynkin.html#T11" target="_self" title="DYNKIN:th.11">Th14</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">Y</font> = <font color="Maroon" title="c3">Y</font> 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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_1"><span class="lab"><font color="Green" title="E26">A1</font></span></a></span>;<br/>
<a NAME="E5:30_1_1"/>
 for <font color="Olive" title="b1">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> holds <br/><font color="Olive" title="b1">z</font> is    <a href="dynkin.html#M1" title="DYNKIN:mode.1">Dynkin_System</a> of <font color="Maroon" title="c1">Omega</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_1"><span class="lab"><font color="Green" title="E26">A1</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">I</font> =  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">Y</font> as    <a href="dynkin.html#M1" title="DYNKIN:mode.1">Dynkin_System</a> of <font color="Maroon" title="c1">Omega</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="dynkin.html#T12" target="_self" title="DYNKIN:th.12">Th15</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">I</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">I</font> &amp; (  for <font color="Olive" title="b1">D</font> being    <a href="dynkin.html#M1" title="DYNKIN:mode.1">Dynkin_System</a> of <font color="Maroon" title="c1">Omega</font>  st <font color="Maroon" title="c2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">D</font> holds <br/><font color="Maroon" title="c5">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">D</font> ) )</span><br/>

<a NAME="E7:30_1_1"/>
 for <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">Y</font> holds  <font color="Maroon" title="c2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_1"><span class="lab"><font color="Green" title="E26">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:30_1_1"/>
<font color="Maroon" title="c2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">I</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dynkin.html#T4" target="_self" title="DYNKIN:th.4">Th7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">D</font> being    <a href="dynkin.html#M1" title="DYNKIN:mode.1">Dynkin_System</a> of <font color="Maroon" title="c1">Omega</font>  st <font color="Maroon" title="c2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">D</font> holds <br/><font color="Maroon" title="c5">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">D</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">D</font> be    <a href="dynkin.html#M1" title="DYNKIN:mode.1">Dynkin_System</a> of <font color="Maroon" title="c1">Omega</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">D</font> implies <font color="Maroon" title="c5">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">D</font> )</span><br/>

<span class="kw">assume </span><a NAME="E9:30_1_1"/>
<font color="Maroon" title="c2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">D</font></span><br/>

<a NAME="E10:30_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_1"><span class="lab"><font color="Green" title="E26">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:30_1_1"/>
<font color="Maroon" title="c5">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#T3" title="SETFAM_1:th.3">SETFAM_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
