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<b>let </b><font color="Maroon">V</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">C</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span>, <font color="Maroon">B</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span>;<br/><b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ;<br/>







<b>assume </b><a NAME="E1:27"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="heyting2.html#K3" target="_self">=&gt;&gt;</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:27"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> </span>)</span> where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> :  <a href="relat_1.html#K1">dom</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> </span>}</span> 
 
;<br/>
<a NAME="E3:27"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> </span>)</span> where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> :  <a href="relat_1.html#K1">dom</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">f</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">A</font>,<font color="Maroon">B</font><b> such that </b><br/><a NAME="E5:27"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span>  &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 ;<br/>
<a NAME="E6:27"/>
<font color="Maroon">f</font> is   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="partfun1.html#T121">PARTFUN1:121</a></i>;<br/>
<b>hence </b><a NAME="E7:27"/>
 ex <font color="Olive">f</font> being  <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> st <br/>( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span>  &amp;  <a href="relset_1.html#K4">dom</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/>


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