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<div><div><a NAME="S28"><span class="kw">scheme  </span><span class="comment"><font color="firebrick">:: FRAENKEL:sch 28</font></span><br/></a><span class="lab"><font color="Green" title="E28">FraenkelFin9</font></span>{ <font color="Maroon">F<sub>1</sub></font>() <span class="kw">-&gt; </span>  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> , <font color="Maroon">F<sub>2</sub></font>() <span class="kw">-&gt; </span>  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> , <font color="Maroon">F<sub>3</sub></font>() <span class="kw">-&gt; </span>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> , <font color="Maroon">P<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] } :<br/><div class="add"><a NAME="E3:34"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() :  ex <font color="Olive" title="b2">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">w</font>,<font color="Olive" title="b1">x</font>] &amp; <font color="Olive" title="b2">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() ) </span> } </span>  is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 </div><span class="kw">provided</span><div class="add"><a NAME="E1:34"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
<font color="Maroon">F<sub>3</sub></font>() is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">and </span><br/><a NAME="E2:34"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
 for <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">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">w</font>,<font color="Olive" title="b2">x</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">w</font>,<font color="Olive" title="b3">y</font>] holds <br/><font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y</font>
 </div></div></div>
