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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">M</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<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#M3">ManySortedFunction</a> of <font color="Maroon">M</font>,<font color="Maroon">M</font>;<br/>







<b>thus </b><a NAME="E1:67"/>
( <font color="Maroon">A</font> <a href="pboole.html#R1">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">F</font> <a href="closure1.html#K1" target="_self">..</a> <font color="Maroon">A</font> <a href="pboole.html#R6">=</a> <font color="Maroon">A</font> implies <font color="Maroon">A</font> <a href="pboole.html#R1">in</a>  <a href="closure1.html#K6" target="_self">MSFixPoints</a> <font color="Maroon">F</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:67_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">A</font> <a href="pboole.html#R1">in</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E2:67_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">F</font> <a href="closure1.html#K1" target="_self">..</a> <font color="Maroon">A</font> <a href="pboole.html#R6">=</a> <font color="Maroon">A</font>
 ;<br/>

<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#D4">PBOOLE:def 4</a><br/>

<b>assume </b><a NAME="E3:67_1"/><i><font color="Green">E37</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">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">M</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="E5:67_1"/>
<font color="Maroon">M</font> <a href="pboole.html#R6">=</a>  <a href="mssubfam.html#K1">doms</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="mssubfam.html#T17">MSSUBFAM:17</a></i>;<br/>
<a NAME="E6:67_1"/><b>then </b>
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="mssubfam.html#T14">MSSUBFAM:14</a></i>;<br/>
<a NAME="E7:67_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="pboole.html#D4">PBOOLE:def 4</a></i>;<br/>
<a NAME="E8:67_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E9:67_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <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="pralg_1.html#D17">PRALG_1:def 17</a></i>;<br/>
<b>hence </b><a NAME="E10:67_1"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="closure1.html#K6" target="_self">MSFixPoints</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:67"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">A</font> <a href="pboole.html#R1">in</a>  <a href="closure1.html#K6" target="_self">MSFixPoints</a> <font color="Maroon">F</font>
 ;<br/>

<b>thus </b><a NAME="E3:67"/>
<font color="Maroon">A</font> <a href="pboole.html#R1">in</a> <font color="Maroon">M</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2"><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#D4">PBOOLE:def 4</a><br/>

<b>assume </b><a NAME="E1:67_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<a NAME="E2:67_2"/><b>then </b>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="closure1.html#K6" target="_self">MSFixPoints</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#D4">PBOOLE:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:67_2"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E5:67_2"/>
<font color="Maroon">M</font> <a href="pboole.html#R6">=</a>  <a href="mssubfam.html#K1">doms</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="mssubfam.html#T17">MSSUBFAM:17</a></i>;<br/>
<b>hence </b><a NAME="E6:67_2"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="mssubfam.html#T14">MSSUBFAM:14</a></i>;<br/>


</div><b>end;</b></div>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_3"><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="E1:67_3"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E2:67_3"/><b>then </b>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="closure1.html#K6" target="_self">MSFixPoints</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#D4">PBOOLE:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:67_3"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E5:67_3"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>hence </b><a NAME="E6:67_3"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="closure1.html#K1" target="_self">..</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <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="pralg_1.html#D17">PRALG_1:def 17</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:67"/>
<font color="Maroon">F</font> <a href="closure1.html#K1" target="_self">..</a> <font color="Maroon">A</font> <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/>


</div>
