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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>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="pboole.html#NM2">ManySortedFunction</a> of <font color="Maroon">I</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:3"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">F</font> is <a href="msualg_3.html#V1">"1-1"</a>
 <b>and </b><br/><a NAME="E2:3"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">A</font> <a href="pboole.html#R2">c=</a>  <a href="extens_1.html#K1">doms</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E3:3"/><i><font color="Green">E48</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</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="3_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="E1:3_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E2:3_1"/><b>then </b>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="extens_1.html#K1">doms</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/><a NAME="E3:3_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>;<br/><a NAME="E4:3_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, , , <a class="ref" href="msualg_3.html#D2">MSUALG_3:def 2</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="equation.html#K1">""</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="pboole.html#K13">.:.:</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="pboole.html#K13">.:.:</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="equation.html#D1">EQUATION:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="pboole.html#D25">PBOOLE:def 25</a></i>
<br/>.= 
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , </i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:3"/>
<font color="Maroon">F</font> <a href="equation.html#K1">""</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pboole.html#K13">.:.:</a> <font color="Maroon">A</font></span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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