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<b>let </b><font color="Maroon" title="c1">J</font>, <font color="Maroon" title="c2">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c4">K</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">J</font>;<br/><b>let </b><font color="Maroon" title="c5">F</font> be   <a href="waybel_5.html#NM1" title="WAYBEL_5:NM.1">DoubleIndexedSet</a> of <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c2">A</font>;<br/><b>let </b><font color="Maroon" title="c6">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>;<br/>









<b>set </b><font color="Maroon" title="c7">fF</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="waybel_5.html#K8" title="WAYBEL_5:func.8">=&gt;</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Maroon" title="c5">F</font>;<br/>
<a NAME="E1:63"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="waybel_5.html#K8" title="WAYBEL_5:func.8">=&gt;</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">J</font>
 
<b>by </b><i><a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">fF</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="waybel_5.html#K8" title="WAYBEL_5:func.8">=&gt;</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Maroon" title="c5">F</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">J</font> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:63"/>
for <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">J</font> holds <br/><font color="Maroon" title="c8">fF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:63_1"/><i><font color="Green" title="E41">A1</font></i>: 
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">J</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c10">j'</font> = <font color="Maroon" title="c9">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">J</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c11">Fj</font> = <font color="Maroon" title="c5">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c10">j'</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c2">A</font> ;<br/>
<a NAME="E4:63_1"/><i><font color="Green" title="E42">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E1:63_1"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<font color="Maroon" title="c8">fF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="waybel_5.html#K8" title="WAYBEL_5:func.8">=&gt;</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_5.html#E1:63_1"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">Fj</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/>
<b>hence </b><a NAME="E6:63_1"/>
<font color="Maroon" title="c8">fF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E4:63_1"><i><font color="Green" title="E42">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:63"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">J</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <font color="Maroon" title="c5">F</font> is   <a href="waybel_5.html#NM1" title="WAYBEL_5:NM.1">DoubleIndexedSet</a> of <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/>


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