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<b>deffunc </b><font color="Maroon">H<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>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon">F<sub>3</sub></font>($2,$1);<br/>
<a NAME="E1:8_1"/><i><font color="Green" title="E3">A3</font></i>: 
for <font color="Olive" title="b1">u</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">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() holds  <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font>)
 
<b>by </b><i><a class="txt" href="fraenkel.html#E2:8"><i><font color="Green" title="E2">A2</font></i></a></i>;<br/>
<a NAME="E2:8_1"/><i><font color="Green" title="E4">A4</font></i>: 
for <font color="Olive" title="b1">u</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">v</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">u</font>,<font color="Olive" title="b2">v</font>] holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font>]
 
<b>by </b><i><a class="txt" href="fraenkel.html#E1:8"><i><font color="Green" title="E1">A1</font></i></a></i>;<br/>
<a NAME="E3:8_1"/><i><font color="Green" title="E5">A5</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">u1</font>,<font color="Olive" title="b2">v1</font>) where <font color="Olive" title="b1">u1</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">v1</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>1</sub></font>[<font color="Olive" title="b1">u1</font>,<font color="Olive" title="b2">v1</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">u2</font>,<font color="Olive" title="b2">v2</font>) where <font color="Olive" title="b1">u2</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">v2</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">u2</font>,<font color="Olive" title="b2">v2</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S2" target="_self" title="FRAENKEL:sch.2">FRAENKEL:sch 2</a>(<a class="txt" href="fraenkel.html#E2:8_1"><i><font color="Green" title="E4">A4</font></i></a>);<br/></i>
<a NAME="E4:8_1"/><i><font color="Green" title="E6">A6</font></i>: 
for <font color="Olive" title="b1">u</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">v</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>2</sub></font>[<font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font>]
 
<b>by </b><i><a class="txt" href="fraenkel.html#E1:8"><i><font color="Green" title="E1">A1</font></i></a></i>;<br/>
<a NAME="E5:8_1"/><i><font color="Green" title="E7">A7</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">u1</font>,<font color="Olive" title="b2">v1</font>) where <font color="Olive" title="b1">u1</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">v1</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">u1</font>,<font color="Olive" title="b2">v1</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">u2</font>,<font color="Olive" title="b2">v2</font>) where <font color="Olive" title="b1">u2</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">v2</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>1</sub></font>[<font color="Olive" title="b1">u2</font>,<font color="Olive" title="b2">v2</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S2" target="_self" title="FRAENKEL:sch.2">FRAENKEL:sch 2</a>(<a class="txt" href="fraenkel.html#E4:8_1"><i><font color="Green" title="E6">A6</font></i></a>);<br/></i>
<a NAME="E6:8_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">u1</font>,<font color="Olive" title="b2">v1</font>) where <font color="Olive" title="b1">u1</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">v1</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">u1</font>,<font color="Olive" title="b2">v1</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">u2</font>,<font color="Olive" title="b2">v2</font>) where <font color="Olive" title="b1">u2</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">v2</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">u2</font>,<font color="Olive" title="b2">v2</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S7" target="_self" title="FRAENKEL:sch.7">FRAENKEL:sch 7</a>(<a class="txt" href="fraenkel.html#E1:8_1"><i><font color="Green" title="E3">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E7:8_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">u1</font>,<font color="Olive" title="b2">v1</font>) where <font color="Olive" title="b1">u1</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">v1</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>1</sub></font>[<font color="Olive" title="b1">u1</font>,<font color="Olive" title="b2">v1</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b2">v2</font>,<font color="Olive" title="b1">u2</font>) where <font color="Olive" title="b1">u2</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">v2</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">u2</font>,<font color="Olive" title="b2">v2</font>] </span>}</span> 
 <b>by </b><i><a class="txt" href="fraenkel.html#E3:8_1"><i><font color="Green" title="E5">A5</font></i></a>, <a class="txt" href="fraenkel.html#E5:8_1"><i><font color="Green" title="E7">A7</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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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