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<span class="kw">defpred </span><font color="Maroon">S<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> ] <span class="kw">means </span><font color="Maroon">P<sub>1</sub></font>[$1,$2];<br/>
<a NAME="E1:181_1"/><span class="lab"><font color="Green" title="E84">A2</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/> ex <font color="Olive" title="b2">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:181"><span class="lab"><font color="Green" title="E83">A1</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c1">f</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon">F<sub>1</sub></font>()<span class="kw"> such that </span><br/><a NAME="E3:181_1"/><span class="lab"><font color="Green" title="E85">A3</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="pboole.html#S3" target="_self" title="PBOOLE:sch.3">PBOOLE:sch 3</a>(<a class="txt" href="#E1:181_1"><span class="lab"><font color="Green" title="E84">A2</font></span></a>);<br/></span>
<span class="kw">take </span>
<font color="Maroon" title="c1">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font>]</span><br/>

<span class="kw">thus </span><a NAME="E4:181_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:181_1"><span class="lab"><font color="Green" title="E85">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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