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<b>let </b><font color="Maroon">I1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">I2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A1</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I1</font>, <font color="Maroon">B1</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I1</font>;<br/><b>let </b><font color="Maroon">A2</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I2</font>, <font color="Maroon">B2</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I2</font>;<br/><b>let </b><font color="Maroon">A</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">I2</font>, <font color="Maroon">B</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">I2</font>;<br/>







<b>assume </b><a NAME="E1:19"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="yellow20.html#K1" target="_self">Intersect</a> <font color="Maroon">A1</font>,<font color="Maroon">A2</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="yellow20.html#K1" target="_self">Intersect</a> <font color="Maroon">B1</font>,<font color="Maroon">B2</font> )
 ;<br/>

<b>let </b><font color="Maroon">F</font> be    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">A1</font>,<font color="Maroon">B1</font>;<br/><b>let </b><font color="Maroon">G</font> be    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">A2</font>,<font color="Maroon">B2</font>;<br/>



<b>assume </b><a NAME="E2:19"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="partfun1.html#R1">tolerates</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">H</font> =  <a href="yellow20.html#K1" target="_self">Intersect</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">I2</font> <b>by </b><i/>;<br/>
<a NAME="E4:19"/><i><font color="Green">E33</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I1</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I2</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I1</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I2</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">I2</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I1</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I2</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">I2</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:19"/>
<font color="Maroon">H</font> is    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_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="E1:19_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">I2</font>
 ;<br/>

<a NAME="E2:19_1"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I1</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I2</font> &amp; <font color="Maroon">A</font> <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">A1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">B</font> <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">B1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">B2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">H</font> <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">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> )
 
<b>by </b><i>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:19_1"/><b>then </b>
( <font color="Maroon">F</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">A1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">B1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">G</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">A2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">B2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="partfun1.html#R1">tolerates</a> <font color="Maroon">G</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>hence </b><a NAME="E4:19_1"/>
<font color="Maroon">H</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">B</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="E6:19"/>
 <a href="yellow20.html#K1" target="_self">Intersect</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> is    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>
 ;<br/>


</div>
