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<b>let </b><font color="Maroon">I</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>, <font color="Maroon">B</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>, <font color="Maroon">C</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/>



<b>assume </b><a NAME="E1:6"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">A</font> <a href="pzfmisc1.html#R1">is_transformable_to</a> <font color="Maroon">B</font>
 ;<br/>

<b>let </b><font color="Maroon">F</font> be    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">G</font> be    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font>;<br/>



<b>reconsider </b><font color="Maroon">GF</font> = <font color="Maroon">G</font> <a href="pboole.html#K12">**</a> <font color="Maroon">F</font> as   <a href="pboole.html#NM2">ManySortedFunction</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="mssubfam.html#T15">MSSUBFAM:15</a></i>;<br/>
<a NAME="E3:6"/>
<font color="Maroon">GF</font> is    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">A</font>,<font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</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="E1:6_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<a NAME="E2:6_1"/><b>then </b><i><font color="Green">E20</font></i>: 
( <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, <a class="ref" href="pzfmisc1.html#D3">PZFMISC1:def 3</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">i</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <b>by </b><i>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Fi</font> = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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">i</font>,<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <b>by </b><i>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<a NAME="E5:6_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">GF</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E6:6_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="pboole.html#K12">**</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Gi</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">Fi</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D24">PBOOLE:def 24</a></i>;<br/>
<b>hence </b><a NAME="E7:6_1"/>
<font color="Maroon">GF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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">i</font>,<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T19">FUNCT_2:19</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:6"/>
<font color="Maroon">G</font> <a href="pboole.html#K12">**</a> <font color="Maroon">F</font> is    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">A</font>,<font color="Maroon">C</font>
 ;<br/>


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