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<b>let </b><font color="Maroon" title="c1">I1</font>, <font color="Maroon" title="c2">I2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">A1</font>, <font color="Maroon" title="c4">B1</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I1</font>;<br/><b>let </b><font color="Maroon" title="c5">A2</font>, <font color="Maroon" title="c6">B2</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c2">I2</font>;<br/><b>let </b><font color="Maroon" title="c7">A</font>, <font color="Maroon" title="c8">B</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">I2</font>;<br/>







<b>assume </b><a NAME="E1:19"/><i><font color="Green" title="E14">A1</font></i>: 
( <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c5">A2</font> &amp; <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <font color="Maroon" title="c4">B1</font>,<font color="Maroon" title="c6">B2</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c9">F</font> be    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c4">B1</font>;<br/><b>let </b><font color="Maroon" title="c10">G</font> be    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c5">A2</font>,<font color="Maroon" title="c6">B2</font>;<br/>



<b>assume </b><a NAME="E2:19"/><i><font color="Green" title="E15">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">F</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">G</font> holds <br/><font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <font color="Maroon" title="c10">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c11">H</font> =  <a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c10">G</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">I2</font> <b>by </b><i><a class="ref" href="yellow20.html#T14" target="_self" title="YELLOW20:th.14">Th14</a></i>;<br/>
<a NAME="E4:19"/><i><font color="Green" title="E16">A3</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I1</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">I2</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I1</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">I2</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">I2</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I1</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">I2</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">I2</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:19"/>
<font color="Maroon" title="c11">H</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">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" title="c12">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a><br/>

<b>assume </b><a NAME="E1:19_1"/>
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">I2</font>
 ;<br/>

<a NAME="E2:19_1"/><b>then </b><i><font color="Green" title="E17">A4</font></i>: 
( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I1</font> &amp; <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I2</font> &amp; <font color="Maroon" title="c7">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> &amp; <font color="Maroon" title="c8">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> &amp; <font color="Maroon" title="c11">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="yellow20.html#E1:19"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="yellow20.html#E4:19"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="yellow20.html#D2" target="_self" title="YELLOW20:def.2">Def2</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:19_1"/><b>then </b>
( <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">A1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c4">B1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c10">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">A2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c6">B2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <font color="Maroon" title="c10">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> )
 
<b>by </b><i><a class="txt" href="yellow20.html#E2:19"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="yellow20.html#E4:19"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/>
<b>hence </b><a NAME="E4:19_1"/>
<font color="Maroon" title="c11">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c7">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c8">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 <b>by </b><i><a class="txt" href="yellow20.html#E2:19_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="yellow20.html#T16" target="_self" title="YELLOW20:th.16">Th16</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:19"/>
 <a href="yellow20.html#K1" title="YELLOW20:func.1">Intersect</a> <font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c10">G</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font>
 ;<br/>


</div>
