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<b>reconsider </b><font color="Maroon" title="c5">G</font> =  <a href="altcat_2.html#K1" title="ALTCAT_2:func.1">~</a> <font color="Maroon" title="c4">F</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ;<br/>
<a NAME="E2:9_1_1"/>
<font color="Maroon" title="c5">G</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of  <a href="altcat_2.html#K1" title="ALTCAT_2:func.1">~</a> <font color="Maroon" title="c2">A</font>, <a href="altcat_2.html#K1" title="ALTCAT_2:func.1">~</a> <font color="Maroon" title="c3">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">ii</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="E1:9_1_1_1"/>
<font color="Maroon" title="c6">ii</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">i1</font>, <font color="Maroon" title="c8">i2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:9_1_1_1"/><i><font color="Green" title="E1">A1</font></i>: 
( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> &amp; <font color="Maroon" title="c8">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> )
 <b>and </b><br/><a NAME="E4:9_1_1_1"/><i><font color="Green" title="E2">A2</font></i>: 
<font color="Maroon" title="c6">ii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></i>;<br/>
<a NAME="E5:9_1_1_1"/><i><font color="Green" title="E3">A3</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c7">i1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E3:9_1_1_1"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E6:9_1_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E7:9_1_1_1"/><b>then </b><i><font color="Green" title="E4">A4</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E5:9_1_1_1"><i><font color="Green" title="E3">A3</font></i></a>, <a class="ref" href="funct_4.html#D2" title="FUNCT_4:def.2">FUNCT_4:def 2</a></i>;<br/>
<a NAME="E8:9_1_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E9:9_1_1_1"/><b>then </b><i><font color="Green" title="E5">A5</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="altcat_2.html#K1" title="ALTCAT_2:func.1">~</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E5:9_1_1_1"><i><font color="Green" title="E3">A3</font></i></a>, <a class="ref" href="funct_4.html#D2" title="FUNCT_4:def.2">FUNCT_4:def 2</a></i>;<br/>
<a NAME="E10:9_1_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E11:9_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">B</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">i2</font>,<font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="altcat_2.html#K1" title="ALTCAT_2:func.1">~</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">i2</font>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E5:9_1_1_1"><i><font color="Green" title="E3">A3</font></i></a>, <a class="ref" href="funct_4.html#D2" title="FUNCT_4:def.2">FUNCT_4:def 2</a></i>;<br/>
<b>hence </b><a NAME="E12:9_1_1_1"/>
<font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">ii</font> is   <a href="relset_1.html#M2" title="RELSET_1:mode.2">M5</a>(<span class="p1">(<span class="default"><a href="altcat_2.html#K1" title="ALTCAT_2:func.1">~</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">ii</font>,<span class="p1">(<span class="default"><a href="altcat_2.html#K1" title="ALTCAT_2:func.1">~</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">ii</font>)
 <b>by </b><i><a class="txt" href="altcat_2.html#E4:9_1_1_1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="altcat_2.html#E5:9_1_1_1"><i><font color="Green" title="E3">A3</font></i></a>, <a class="txt" href="altcat_2.html#E7:9_1_1_1"><i><font color="Green" title="E4">A4</font></i></a>, <a class="txt" href="altcat_2.html#E9:9_1_1_1"><i><font color="Green" title="E5">A5</font></i></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:9_1_1"/>
 <a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <font color="Maroon" title="c4">F</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of  <a href="altcat_2.html#K1" title="ALTCAT_2:func.1">~</a> <font color="Maroon" title="c2">A</font>, <a href="altcat_2.html#K1" title="ALTCAT_2:func.1">~</a> <font color="Maroon" title="c3">B</font>
 ;<br/>


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