<?xml version="1.0"?>
<div class="add">

<b>consider </b><font color="Maroon" title="c2">M</font> being    <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">A</font>,<font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><b> such that </b><br/><a NAME="E2:40_1_1"/><i><font color="Green" title="E26">A1</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><font color="Maroon" title="c2">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>
 <b>from </b><i><a class="ref" href="altcat_1.html#S1" target="_self" title="ALTCAT_1:sch.1">ALTCAT_1:sch 1</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b> <a href="altcat_1.html#V7" title="ALTCAT_1:attr.7">compositional</a> <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> $2,$3</span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> $1,$2</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> =  <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> $1,$2</span>)</span>,<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> $2,$3</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c3">c</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span><b> such that </b><br/><a NAME="E4:40_1_1"/><i><font color="Green" title="E27">A2</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font>, <font color="Olive" title="b3">k</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><font color="Maroon" title="c3">c</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">k</font>)
 <b>from </b><i><a class="ref" href="altcat_1.html#S3" target="_self" title="ALTCAT_1:sch.3">ALTCAT_1:sch 3</a>();<br/></i>
<a NAME="E5:40_1_1"/>
<font color="Maroon" title="c3">c</font> is <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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="funcop_1.html#D6" title="FUNCOP_1:def.6">FUNCOP_1:def 6</a><br/>

<b>assume </b><a NAME="E1:40_1_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">c</font>
 ;<br/>

<a NAME="E2:40_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c6">x2</font>, <font color="Maroon" title="c7">x3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:40_1_1_1"/><i><font color="Green" title="E28">A3</font></i>: 
( <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c7">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 <b>and </b><br/><a NAME="E5:40_1_1_1"/><i><font color="Green" title="E29">A4</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 <b>by </b><i><a class="ref" href="mcart_1.html#T72" title="MCART_1:th.72">MCART_1:72</a></i>;<br/>
<font color="Maroon" title="c3">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> = 
<font color="Maroon" title="c3">c</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font>
<b>by </b><i><a class="txt" href="altcat_1.html#E5:40_1_1_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>
<br/>.= 
 <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font></span>)</span>,<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">x3</font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_1.html#E4:40_1_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="altcat_1.html#E4:40_1_1_1"><i><font color="Green" title="E28">A3</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E7:40_1_1_1"/>
<font color="Maroon" title="c3">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">c</font> = <font color="Maroon" title="c3">c</font> as   <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:40_1_1_2"/>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">x2</font>, <font color="Maroon" title="c8">x3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:40_1_1_2"/><i><font color="Green" title="E28">A5</font></i>: 
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c8">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 <b>and </b><br/><a NAME="E4:40_1_1_2"/><i><font color="Green" title="E29">A6</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c8">x3</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 <b>by </b><i><a class="ref" href="mcart_1.html#T72" title="MCART_1:th.72">MCART_1:72</a></i>;<br/><a NAME="E5:40_1_1_2"/><i><font color="Green" title="E30">A7</font></i>: 
<font color="Maroon" title="c2">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">x2</font>
 <b>by </b><i><a class="txt" href="altcat_1.html#E2:40_1_1"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="altcat_1.html#E3:40_1_1_2"><i><font color="Green" title="E28">A5</font></i></a></i>;<br/><a NAME="E6:40_1_1_2"/><i><font color="Green" title="E31">A8</font></i>: 
<font color="Maroon" title="c2">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c8">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c8">x3</font>
 <b>by </b><i><a class="txt" href="altcat_1.html#E2:40_1_1"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="altcat_1.html#E3:40_1_1_2"><i><font color="Green" title="E28">A5</font></i></a></i>;<br/><a NAME="E7:40_1_1_2"/><i><font color="Green" title="E32">A9</font></i>: 
<font color="Maroon" title="c2">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c8">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c8">x3</font>
 <b>by </b><i><a class="txt" href="altcat_1.html#E2:40_1_1"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="altcat_1.html#E3:40_1_1_2"><i><font color="Green" title="E28">A5</font></i></a></i>;<br/><i><font color="Green" title="E33">A10</font></i>: <font color="Maroon" title="c4">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> = 
<font color="Maroon" title="c4">c</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c8">x3</font>
<b>by </b><i><a class="txt" href="altcat_1.html#E4:40_1_1_2"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>
<br/>.= 
 <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">x2</font></span>)</span>,<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c8">x3</font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_1.html#E4:40_1_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="altcat_1.html#E3:40_1_1_2"><i><font color="Green" title="E28">A5</font></i></a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c9">ci</font> = <font color="Maroon" title="c4">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/><a NAME="E10:40_1_1_2"/><i><font color="Green" title="E34">A11</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">ci</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"><span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c8">x3</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">x2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="altcat_1.html#E8:40_1_1_2"><i><font color="Green" title="E33">A10</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><i><font color="Green" title="E35">A12</font></i>: <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c8">x3</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">x2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> = 
<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c2">M</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c8">x3</font>
<b>by </b><i><a class="txt" href="altcat_1.html#E3:40_1_1_2"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="altcat_1.html#E5:40_1_1_2"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="altcat_1.html#E6:40_1_1_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="altcat_1.html#D6" target="_self" title="ALTCAT_1:def.6">Def6</a></i>
<br/>.= 
<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c2">M</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
<b>by </b><i><a class="txt" href="altcat_1.html#E4:40_1_1_2"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>
;<br/><i><font color="Green" title="E36">A13</font></i>: <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c2">M</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> = 
<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c2">M</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c8">x3</font>
<b>by </b><i><a class="txt" href="altcat_1.html#E4:40_1_1_2"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c2">M</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c8">x3</font>
<b>by </b><i><a class="txt" href="altcat_1.html#E3:40_1_1_2"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="altcat_1.html#D5" target="_self" title="ALTCAT_1:def.5">Def5</a></i>
;<br/><a NAME="E13:40_1_1_2"/><b>then </b><i><font color="Green" title="E37">A14</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">ci</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c2">M</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="altcat_1.html#E7:40_1_1_2"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="altcat_1.html#E8:40_1_1_2"><i><font color="Green" title="E33">A10</font></i></a>, <a class="ref" href="altcat_1.html#T14" target="_self" title="ALTCAT_1:th.14">Th14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1_1_2_4"/>
<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c2">M</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>then consider </b><font color="Maroon" title="c10">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:40_1_1_2_4"/><i><font color="Green" title="E38">A15</font></i>: 
<font color="Maroon" title="c10">j</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"><span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c8">x3</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">x2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="altcat_1.html#E11:40_1_1_2"><i><font color="Green" title="E35">A12</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">j1</font>, <font color="Maroon" title="c12">j2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:40_1_1_2_4"/><i><font color="Green" title="E39">A16</font></i>: 
<font color="Maroon" title="c11">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c8">x3</font>
 <b>and </b><br/><a NAME="E6:40_1_1_2_4"/><i><font color="Green" title="E40">A17</font></i>: 
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">x2</font>
 <b>and </b><br/><a NAME="E7:40_1_1_2_4"/>
<font color="Maroon" title="c10">j</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="c11">j1</font>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="altcat_1.html#E3:40_1_1_2_4"><i><font color="Green" title="E38">A15</font></i></a>, <a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">j1</font> = <font color="Maroon" title="c11">j1</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c8">x3</font> <b>by </b><i><a class="txt" href="altcat_1.html#E5:40_1_1_2_4"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">j2</font> = <font color="Maroon" title="c12">j2</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">x2</font> <b>by </b><i><a class="txt" href="altcat_1.html#E6:40_1_1_2_4"><i><font color="Green" title="E40">A17</font></i></a>, <a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></i>;<br/><a NAME="E10:40_1_1_2_4"/>
<font color="Maroon" title="c13">j1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c14">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c8">x3</font>
 <b>by </b><i><a class="txt" href="altcat_1.html#E5:40_1_1_2_4"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="altcat_1.html#E6:40_1_1_2_4"><i><font color="Green" title="E40">A17</font></i></a>, <a class="ref" href="altcat_1.html#T4" target="_self" title="ALTCAT_1:th.4">Th4</a></i>;<br/><b>hence </b><a NAME="E11:40_1_1_2_4"/>
<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c2">M</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="altcat_1.html#E2:40_1_1"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="altcat_1.html#E3:40_1_1_2"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="altcat_1.html#E12:40_1_1_2"><i><font color="Green" title="E36">A13</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E15:40_1_1_2"/>
<font color="Maroon" title="c4">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c2">M</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c2">M</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="altcat_1.html#E10:40_1_1_2"><i><font color="Green" title="E34">A11</font></i></a>, <a class="txt" href="altcat_1.html#E11:40_1_1_2"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="altcat_1.html#E13:40_1_1_2"><i><font color="Green" title="E37">A14</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">c</font> = <font color="Maroon" title="c4">c</font> as   <a href="altcat_1.html#NM3" title="ALTCAT_1:NM.3">BinComp</a> of <font color="Maroon" title="c2">M</font> <b>by </b><i><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">C</font> =  <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c5">c</font> #);<br/>
<a NAME="E9:40_1_1"/>
 <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c5">c</font> #) is <a href="altcat_1.html#V10" title="ALTCAT_1:attr.10">pseudo-functional</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">o1</font>, <font color="Maroon" title="c8">o2</font>, <font color="Maroon" title="c9">o3</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c5">c</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D15" target="_self" title="ALTCAT_1:def.15">ALTCAT_1:def 15</a><br/>

<a NAME="E1:40_1_1_3"/><i><font color="Green" title="E28">A18</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>
 
<b>by </b><i><a class="txt" href="altcat_1.html#E2:40_1_1"><i><font color="Green" title="E26">A1</font></i></a></i>;<br/>
<a NAME="E2:40_1_1_3"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font>
 
<b>by </b><i><a class="txt" href="altcat_1.html#E2:40_1_1"><i><font color="Green" title="E26">A1</font></i></a></i>;<br/>
<a NAME="E3:40_1_1_3"/><b>then </b><i><font color="Green" title="E29">A19</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font></span>)</span></span>)</span> <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"><span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="altcat_1.html#E1:40_1_1_3"><i><font color="Green" title="E28">A18</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>thus </b>the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c5">c</font> #) <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font> = 
 <a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font></span>)</span>,<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_1.html#E4:40_1_1"><i><font color="Green" title="E27">A2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="altcat_1.html#K6" title="ALTCAT_1:func.6">FuncComp</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">o2</font>,<font color="Maroon" title="c9">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">o1</font>,<font color="Maroon" title="c8">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="txt" href="altcat_1.html#E3:40_1_1_3"><i><font color="Green" title="E29">A19</font></i></a>, <a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a></i>
;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">C</font> =  <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c5">c</font> #) as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="altcat_1.html#V6" title="ALTCAT_1:attr.6">strict</a> <a href="altcat_1.html#V10" title="ALTCAT_1:attr.10">pseudo-functional</a>  <a href="altcat_1.html#L2" title="ALTCAT_1:struct.2">AltCatStr</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c7">C</font>
;<br/>

<b>thus </b><a NAME="E11:40_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c8">a1</font>, <font color="Maroon" title="c9">a2</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c7">C</font>;<br/>

<b>thus </b><a NAME="E12:40_1_1"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">a1</font>,<font color="Maroon" title="c9">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c8">a1</font>,<font color="Maroon" title="c9">a2</font>
 <b>by </b><i><a class="txt" href="altcat_1.html#E2:40_1_1"><i><font color="Green" title="E26">A1</font></i></a></i>;<br/>


</div>
