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<b>let </b><font color="Maroon">I1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">I2</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">I3</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">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">I1</font>,<font color="Maroon">I2</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">I2</font>,<font color="Maroon">I3</font>;<br/><b>let </b><font color="Maroon">A</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I1</font>;<br/><b>let </b><font color="Maroon">B</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I2</font>;<br/><b>let </b><font color="Maroon">C</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I3</font>;<br/><b>let </b><font color="Maroon">F</font> be    <a href="functor0.html#M1" target="_self">MSUnTrans</a> of <font color="Maroon">f</font>,<font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">G</font> be    <a href="functor0.html#M1" target="_self">MSUnTrans</a> of <font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>,<font color="Maroon">B</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font>,<font color="Maroon">C</font>;<br/>

















<b>assume </b><a NAME="E1:30"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">ii</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">ii</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">ii</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; not <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">ii</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="pboole.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">ii</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</font> as    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">B</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font>,<font color="Maroon">C</font> <a href="pboole.html#K7">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font> <b>by </b><i/>;<br/>
<a NAME="E4:30"/><i><font color="Green">E23</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I1</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I1</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<i><font color="Green">E26</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="altcat_2.html#K3">**</a> <font color="Maroon">F</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D24">PBOOLE:def 24</a></i>
<br/>.= 
<font color="Maroon">I1</font>
<b>by </b><i/>
;<br/>
<b>reconsider </b><font color="Maroon">GF</font> = <font color="Maroon">G</font> <a href="altcat_2.html#K3">**</a> <font color="Maroon">F</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I1</font> ;<br/>
<a NAME="E7:30"/>
<font color="Maroon">GF</font> is    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">A</font>,<font color="Maroon">C</font> <a href="pboole.html#K7">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">ii</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a><br/>

<b>assume </b><a NAME="E1:30_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">ii</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">Fi</font> = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ii</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ii</font>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ii</font> <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Gi</font> = <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ii</font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ii</font>,<span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="pboole.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ii</font> <b>by </b><i><a class="txt" href="functor0.html#E5:30"><i><font color="Green">E26</font></i></a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<a NAME="E4:30_2"/>
(  not <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ii</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ii</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="pboole.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ii</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="txt" href="functor0.html#E4:30"><i><font color="Green">E23</font></i></a>, <a class="txt" href="functor0.html#E5:30"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E5:30_2"/><b>then </b>
<font color="Maroon">Gi</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">Fi</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ii</font>,<span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="pboole.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ii</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T19">FUNCT_2:19</a></i>;<br/>
<b>hence </b><a NAME="E6:30_2"/>
<font color="Maroon">GF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ii</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ii</font>,<span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="pboole.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ii</font>
 <b>by </b><i>, <a class="txt" href="functor0.html#E5:30"><i><font color="Green">E26</font></i></a>, <a class="ref" href="pboole.html#D24">PBOOLE:def 24</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:30"/>
<font color="Maroon">G</font> <a href="pboole.html#K12">**</a> <font color="Maroon">F</font> is    <a href="functor0.html#M1" target="_self">MSUnTrans</a> of <font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>,<font color="Maroon">A</font>,<font color="Maroon">C</font>
 <b>by </b><i/>;<br/>


</div>
