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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> , <font color="Maroon">B</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">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/>







<i><font color="Green">E34</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><span class="p3">(<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="msualg_3.html#K3">**</a> <font color="Maroon">F</font></span>)</span></span>)</span> = 
 <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="msualg_3.html#K3">**</a> <font color="Maroon">F</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_6.html#T89">FUNCT_6:89</a></i>
<br/>.= 
<font color="Maroon">J</font>
<b>by </b><i><a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>
;<br/>
<i><font color="Green">E36</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
<b>by </b><i><a class="ref" href="funct_6.html#T89">FUNCT_6:89</a></i>
<br/>.= 
<font color="Maroon">J</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:65_3"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</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">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> ;<br/><a NAME="E3:65_3"/><i><font color="Green">E39</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E4:65_3"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E5:65_3"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <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="msualg_3.html#K3">**</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_5.html#E5"><i><font color="Green">Lemma42</font></i></a>, , <a class="ref" href="funct_6.html#T89">FUNCT_6:89</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><span class="p3">(<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="msualg_3.html#K3">**</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="msualg_3.html#K3">**</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>
;<br/><b>hence </b><a NAME="E7:65_3"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><span class="p3">(<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="msualg_3.html#K3">**</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:65"/>
 <a href="funct_6.html#K2">doms</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="msualg_3.html#K3">**</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E5"><i><font color="Green">Lemma42</font></i></a>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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