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<b>assume </b><a NAME="E1:21_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">y</font>) where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">y</font>] &amp; not <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">y</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() ) </span>}</span>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon">F<sub>2</sub></font>()
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:21_1"/><i><font color="Green" title="E4">A1</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">y</font>) where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">y</font>] &amp; not <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">y</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() ) </span>}</span>  &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:21_1"/>
 ex <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">F<sub>1</sub></font>() st <br/>( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">y</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">y</font>] &amp; not <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">y</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() )
 
<b>by </b><i><a class="txt" href="fraenkel.html#E3:21_1"><i><font color="Green" title="E4">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:21_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fraenkel.html#E3:21_1"><i><font color="Green" title="E4">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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