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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="relat_1.html#V2">V2</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>, <font color="Maroon">B</font> be  <a href="relat_1.html#V2">V2</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>, <font color="Maroon">C</font> be  <a href="relat_1.html#V2">V2</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</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>assume </b><b>that </b><br/><a NAME="E1:22"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">F</font> is <a href="msualg_3.html#V1">"1-1"</a> &amp; <font color="Maroon">F</font> is <a href="msualg_3.html#V2">"onto"</a> )
 <b>and </b><br/><a NAME="E2:22"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">G</font> is <a href="msualg_3.html#V1">"1-1"</a> &amp; <font color="Maroon">G</font> is <a href="msualg_3.html#V2">"onto"</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:22_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><b>then reconsider </b><font color="Maroon">f</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/><b>reconsider </b><font color="Maroon">g</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/><a NAME="E4:22_1"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> 
 <b>by </b><i>, , <a class="ref" href="msualg_3.html#D5">MSUALG_3:def 5</a></i>;<br/><a NAME="E5:22_1"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K2">"</a> 
 <b>by </b><i>, , <a class="ref" href="msualg_3.html#D5">MSUALG_3:def 5</a></i>;<br/><a NAME="E6:22_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">G</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">F</font> is <a href="msualg_3.html#V1">"1-1"</a> &amp; <font color="Maroon">G</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">F</font> is <a href="msualg_3.html#V2">"onto"</a> )
 <b>by </b><i>, , , </i>;<br/><a NAME="E7:22_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, , <a class="ref" href="msualg_3.html#T1">MSUALG_3:1</a></i>;<br/><a NAME="E8:22_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, , <a class="ref" href="msualg_3.html#T1">MSUALG_3:1</a></i>;<br/><a NAME="E9:22_1"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="msualg_3.html#K3">**</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">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>;<br/><a NAME="E10:22_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">F</font></span>)</span> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</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> <a href="funct_1.html#K2">"</a> 
 <b>by </b><i>, , <a class="ref" href="msualg_3.html#D5">MSUALG_3:def 5</a></i>;<br/><a NAME="E11:22_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <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="msualg_3.html#D3">MSUALG_3:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">ff</font> = <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K4">""</a> </span>)</span> <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">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <b>by </b><i>, , <a class="ref" href="funct_2.html#T31">FUNCT_2:31</a></i>;<br/><a NAME="E13:22_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <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="msualg_3.html#D3">MSUALG_3:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">gg</font> = <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="msualg_3.html#K4">""</a> </span>)</span> <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">C</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="funct_2.html#T31">FUNCT_2:31</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="msualg_3.html#K4">""</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">ff</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">gg</font>
<b>by </b><i>, <a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>
<br/>.= 
<font color="Maroon">ff</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2">"</a> </span>)</span>
<b>by </b><i>, , <a class="ref" href="msualg_3.html#D5">MSUALG_3:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2">"</a> </span>)</span>
<b>by </b><i>, , <a class="ref" href="msualg_3.html#D5">MSUALG_3:def 5</a></i>
;<br/><b>hence </b><a NAME="E16:22_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">F</font></span>)</span> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="msualg_3.html#K4">""</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T66">FUNCT_1:66</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:22"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">F</font></span>)</span> <a href="msualg_3.html#K4">""</a>  <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="msualg_3.html#K4">""</a> </span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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