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<b>let </b><font color="Maroon">J</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">K</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="waybel_5.html#NM1">DoubleIndexedSet</a> of <font color="Maroon">K</font>,<font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/>









<b>set </b><font color="Maroon">fF</font> = <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel_5.html#K8" target="_self">=&gt;</a> <font color="Maroon">f</font></span>)</span> <a href="pboole.html#K12">**</a> <font color="Maroon">F</font>;<br/>
<a NAME="E1:64"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="waybel_5.html#K8" target="_self">=&gt;</a> <font color="Maroon">f</font></span>)</span> <a href="pboole.html#K12">**</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font>
 
<b>by </b><i><a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">fF</font> = <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel_5.html#K8" target="_self">=&gt;</a> <font color="Maroon">f</font></span>)</span> <a href="pboole.html#K12">**</a> <font color="Maroon">F</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:64"/>
for <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> holds <br/><font color="Maroon">fF</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:64_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">j'</font> = <font color="Maroon">j</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> ;<br/>
<b>reconsider </b><font color="Maroon">Fj</font> = <font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j'</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">A</font> ;<br/>
<a NAME="E4:64_1"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<font color="Maroon">fF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="waybel_5.html#K8" target="_self">=&gt;</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, <a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">Fj</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>
<b>hence </b><a NAME="E6:64_1"/>
<font color="Maroon">fF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E5"><i><font color="Green">Lemma42</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:64"/>
<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">f</font></span>)</span> <a href="pboole.html#K12">**</a> <font color="Maroon">F</font> is   <a href="waybel_5.html#NM1">DoubleIndexedSet</a> of <font color="Maroon">K</font>,<font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>


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