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<b>thus </b><a NAME="E1:22_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>) where <font color="Olive" title="b1">s</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="Olive" title="b2">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">s'</font>,<font color="Maroon">F<sub>4</sub></font>()) where <font color="Olive" title="b1">s'</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">s'</font>,<font color="Maroon">F<sub>4</sub></font>()] </span>}</span> 
  <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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">s'</font>,<font color="Maroon">F<sub>4</sub></font>()) where <font color="Olive" title="b1">s'</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">s'</font>,<font color="Maroon">F<sub>4</sub></font>()] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>) where <font color="Olive" title="b1">s</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="Olive" title="b2">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>] </span>}</span> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">x</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <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">s</font>,<font color="Olive" title="b2">t</font>) where <font color="Olive" title="b1">s</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="Olive" title="b2">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>] </span>}</span>  or <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">s'</font>,<font color="Maroon">F<sub>4</sub></font>()) where <font color="Olive" title="b1">s'</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">s'</font>,<font color="Maroon">F<sub>4</sub></font>()] </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:22_1_1"/>
<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">s</font>,<font color="Olive" title="b2">t</font>) where <font color="Olive" title="b1">s</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="Olive" title="b2">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>] </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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">s'</font>,<font color="Maroon">F<sub>4</sub></font>()) where <font color="Olive" title="b1">s'</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">s'</font>,<font color="Maroon">F<sub>4</sub></font>()] </span>}</span> </span><br/>

<b>then consider </b><font color="Maroon" title="c2">s</font> being    <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" title="c3">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E3:22_1_1"/><i><font color="Green" title="E5">A2</font></i>: 
<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="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">t</font>)
 <b>and </b><br/><a NAME="E4:22_1_1"/><i><font color="Green" title="E6">A3</font></i>: 
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">t</font>]
 ;<br/>
<a NAME="E5:22_1_1"/>
( <font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">t</font>] )
 
<b>by </b><i><a class="txt" href="fraenkel.html#E1:22"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="fraenkel.html#E4:22_1_1"><i><font color="Green" title="E6">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:22_1_1"/>
<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">s'</font>,<font color="Maroon">F<sub>4</sub></font>()) where <font color="Olive" title="b1">s'</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">s'</font>,<font color="Maroon">F<sub>4</sub></font>()] </span>}</span> 
 <b>by </b><i><a class="txt" href="fraenkel.html#E3:22_1_1"><i><font color="Green" title="E5">A2</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/>


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

<b>let </b><font color="Maroon" title="c1">x</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <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">s'</font>,<font color="Maroon">F<sub>4</sub></font>()) where <font color="Olive" title="b1">s'</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">s'</font>,<font color="Maroon">F<sub>4</sub></font>()] </span>}</span>  or <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">s</font>,<font color="Olive" title="b2">t</font>) where <font color="Olive" title="b1">s</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="Olive" title="b2">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>] </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E2:22_1"/>
<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">s'</font>,<font color="Maroon">F<sub>4</sub></font>()) where <font color="Olive" title="b1">s'</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">s'</font>,<font color="Maroon">F<sub>4</sub></font>()] </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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">s</font>,<font color="Olive" title="b2">t</font>) where <font color="Olive" title="b1">s</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="Olive" title="b2">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>] </span>}</span> </span><br/>

<b>then consider </b><font color="Maroon" title="c2">s'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E4:22_1"/><i><font color="Green" title="E5">A4</font></i>: 
<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="Maroon" title="c2">s'</font>,<font color="Maroon">F<sub>4</sub></font>())
 <b>and </b><br/><a NAME="E5:22_1"/><i><font color="Green" title="E6">A5</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">s'</font>,<font color="Maroon">F<sub>4</sub></font>()]
 ;<br/>
<a NAME="E6:22_1"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c2">s'</font>,<font color="Maroon">F<sub>4</sub></font>()]
 
<b>by </b><i><a class="txt" href="fraenkel.html#E1:22"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="fraenkel.html#E5:22_1"><i><font color="Green" title="E6">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:22_1"/>
<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">s</font>,<font color="Olive" title="b2">t</font>) where <font color="Olive" title="b1">s</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="Olive" title="b2">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>] </span>}</span> 
 <b>by </b><i><a class="txt" href="fraenkel.html#E4:22_1"><i><font color="Green" title="E5">A4</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/>


</div>
