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

<b>consider </b><font color="Maroon">M</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E2:36_1"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">M</font> <a href="binop_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">i</font>,<font color="Olive">j</font>)
 <b>from </b><i><a class="ref" href="altcat_1.html#S2">ALTCAT_1:sch 2</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>3</sub></font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>)</span><a href="zfmisc_1.html#K2">:]</a></span> =  <a href="altcat_1.html#K6">FuncComp</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>3</sub></font>);<br/>
<b>consider </b><font color="Maroon">c</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K3">:]</a></span><b> such that </b><br/><a NAME="E4:36_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font>, <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">c</font> <a href="multop_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Olive">j</font>,<font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>,<font color="Olive">j</font>,<font color="Olive">k</font>)
 <b>from </b><i><a class="ref" href="altcat_1.html#S4">ALTCAT_1:sch 4</a>();<br/></i>
<a NAME="E5:36_1"/>
<font color="Maroon">c</font> is <a href="funcop_1.html#V1">Function-yielding</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_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="funcop_1.html#D6">FUNCOP_1:def 6</a><br/>

<b>assume </b><a NAME="E1:36_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c</font>
 ;<br/>

<a NAME="E2:36_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K3">:]</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x3</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:36_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">x3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() )
 <b>and </b><br/><a NAME="E5:36_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="mcart_1.html#K3">]</a></span>
 <b>by </b><i><a class="ref" href="mcart_1.html#T72">MCART_1:72</a></i>;<br/>
<font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">c</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font>
<b>by </b><i>, <a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
<br/>.= 
 <a href="altcat_1.html#K6">FuncComp</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x1</font>,<font color="Maroon">x2</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x2</font>,<font color="Maroon">x3</font>)
<b>by </b><i><a class="txt" href="orders_3.html#E2:36_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="orders_3.html#D2" target="_self">Def2</a></i>
;<br/>
<b>hence </b><a NAME="E7:36_1_1"/>
<font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="funct_1.html#NM1">Function</a>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c</font> as   <a href="pboole.html#NM2">ManySortedFunction</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K3">:]</a></span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2"><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="E1:36_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K3">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x3</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_1_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">x3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() )
 <b>and </b><br/><a NAME="E4:36_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="mcart_1.html#K3">]</a></span>
 <b>by </b><i><a class="ref" href="mcart_1.html#T72">MCART_1:72</a></i>;<br/><a NAME="E5:36_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">M</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x1</font>,<font color="Maroon">x2</font>)
 <b>by </b><i><a class="txt" href="orders_3.html#E1:36"><i><font color="Green">E20</font></i></a>, <a class="ref" href="orders_3.html#D2" target="_self">Def2</a></i>;<br/><i><font color="Green">E46</font></i>: <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">c</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font>
<b>by </b><i>, <a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
<br/>.= 
 <a href="altcat_1.html#K6">FuncComp</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x1</font>,<font color="Maroon">x2</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x2</font>,<font color="Maroon">x3</font>)
<b>by </b><i><a class="txt" href="orders_3.html#E2:36_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="orders_3.html#D2" target="_self">Def2</a></i>
;<br/><b>then reconsider </b><font color="Maroon">ci</font> = <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/><a NAME="E8:36_1_2"/><i><font color="Green">E47</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">ci</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x2</font>,<font color="Maroon">x3</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x1</font>,<font color="Maroon">x2</font>)</span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><i><font color="Green">E48</font></i>: <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x2</font>,<font color="Maroon">x3</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x1</font>,<font color="Maroon">x2</font>)</span><a href="zfmisc_1.html#K2">:]</a></span> = 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x2</font>,<font color="Maroon">x3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="txt" href="orders_3.html#E1:36"><i><font color="Green">E20</font></i></a>, <a class="ref" href="orders_3.html#D2" target="_self">Def2</a>, <a class="ref" href="orders_3.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font>
<b>by </b><i><a class="ref" href="orders_3.html#D2" target="_self">Def2</a>, <a class="ref" href="altcat_1.html#D6">ALTCAT_1:def 6</a></i>
<br/>.= 
<span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
;<br/><a NAME="E10:36_1_2"/><i><font color="Green">E49</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K6">FuncComp</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x1</font>,<font color="Maroon">x2</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x2</font>,<font color="Maroon">x3</font>)</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x1</font>,<font color="Maroon">x3</font>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2_3"><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="tarski.html#D3">TARSKI:def 3</a><br/>

<b>set </b><font color="Maroon">F</font> =  <a href="altcat_1.html#K6">FuncComp</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x1</font>,<font color="Maroon">x2</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x2</font>,<font color="Maroon">x3</font>);<br/>
<b>assume </b><a NAME="E1:36_1_2_3"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K6">FuncComp</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x1</font>,<font color="Maroon">x2</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x2</font>,<font color="Maroon">x3</font>)</span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">j</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_1_2_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K6">FuncComp</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x1</font>,<font color="Maroon">x2</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x2</font>,<font color="Maroon">x3</font>)</span>)</span>
 <b>and </b><br/><a NAME="E4:36_1_2_3"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K6">FuncComp</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x1</font>,<font color="Maroon">x2</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x2</font>,<font color="Maroon">x3</font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:36_1_2_3"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E7:36_1_2_3"/><i><font color="Green">E109</font></i>: 
<span class="p1">(<span class="default"><a href="altcat_1.html#K6">FuncComp</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x1</font>,<font color="Maroon">x2</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x2</font>,<font color="Maroon">x3</font>)</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="altcat_1.html#D11">ALTCAT_1:def 11</a></i>;<br/>
<a NAME="E8:36_1_2_3"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K6">FuncComp</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x1</font>,<font color="Maroon">x2</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x2</font>,<font color="Maroon">x3</font>)</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x2</font>,<font color="Maroon">x3</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x1</font>,<font color="Maroon">x2</font>)</span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E9:36_1_2_3"/><b>then </b>
( <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x2</font>,<font color="Maroon">x3</font>) &amp; <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x1</font>,<font color="Maroon">x2</font>) )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E10:36_1_2_3"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x1</font>,<font color="Maroon">x3</font>)
 <b>by </b><i><a class="txt" href="orders_3.html#E3"><i><font color="Green">Lemma19</font></i></a>, <a class="ref" href="orders_3.html#D2" target="_self">Def2</a>, <a class="txt" href="orders_3.html#E4:36_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="orders_3.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div><i><font color="Green">E110</font></i>: <span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">M</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">M</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font>
<b>by </b><i>, <a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
<br/>.= 
<font color="Maroon">M</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x3</font>
<b>by </b><i><a class="ref" href="orders_3.html#D2" target="_self">Def2</a>, <a class="ref" href="altcat_1.html#D5">ALTCAT_1:def 5</a></i>
;<br/><a NAME="E12:36_1_2"/><b>then </b><i><font color="Green">E111</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">ci</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">M</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="orders_3.html#E1:36"><i><font color="Green">E20</font></i></a>, <a class="ref" href="orders_3.html#D2" target="_self">Def2</a>, , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_2_5"/>
<span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then consider </b><font color="Maroon">j</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_1_2_5"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x2</font>,<font color="Maroon">x3</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x1</font>,<font color="Maroon">x2</font>)</span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>consider </b><font color="Maroon">j1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">j2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:36_1_2_5"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x2</font>,<font color="Maroon">x3</font>)
 <b>and </b><br/><a NAME="E6:36_1_2_5"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">j2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x1</font>,<font color="Maroon">x2</font>)
 <b>and </b><br/><a NAME="E7:36_1_2_5"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j1</font>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/><b>reconsider </b><font color="Maroon">j1</font> = <font color="Maroon">j1</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="orders_3.html#T4" target="_self">Th4</a>, <a class="ref" href="fraenkel.html#D1">FRAENKEL:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">j2</font> = <font color="Maroon">j2</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i>, <a class="ref" href="fraenkel.html#D1">FRAENKEL:def 1</a></i>;<br/><a NAME="E10:36_1_2_5"/>
<font color="Maroon">j1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">j2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">x1</font>,<font color="Maroon">x3</font>)
 <b>by </b><i><a class="txt" href="orders_3.html#E3"><i><font color="Green">Lemma19</font></i></a>, <a class="ref" href="orders_3.html#D2" target="_self">Def2</a>, <a class="ref" href="orders_3.html#T4" target="_self">Th4</a>, </i>;<br/><b>hence </b><a NAME="E11:36_1_2_5"/>
<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">M</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="orders_3.html#E1:36"><i><font color="Green">E20</font></i></a>, <a class="ref" href="orders_3.html#D2" target="_self">Def2</a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E14:36_1_2"/>
<font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">M</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="txt" href="orders_3.html#E4:36_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c</font> as   <a href="altcat_1.html#NM3">BinComp</a> of <font color="Maroon">M</font> <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<b>set </b><font color="Maroon">C</font> =  <a href="altcat_1.html#G2">AltCatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">M</font>,<font color="Maroon">c</font> #);<br/>
<b>take </b>
 <a href="altcat_1.html#G2">AltCatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">M</font>,<font color="Maroon">c</font> #)
;<br/>

<b>thus </b><a NAME="E9:36_1"/>
the <a href="struct_0.html#U1">carrier</a> of <a href="altcat_1.html#G2">AltCatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">M</font>,<font color="Maroon">c</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>thus </b><a NAME="E10:36_1"/>
the <a href="altcat_1.html#U1">Arrows</a> of <a href="altcat_1.html#G2">AltCatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">M</font>,<font color="Maroon">c</font> #) <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">i</font>,<font color="Maroon">j</font>)
 <b>by </b><i><a class="txt" href="orders_3.html#E1:36"><i><font color="Green">E20</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>thus </b><a NAME="E11:36_1"/>
the <a href="altcat_1.html#U2">Comp</a> of <a href="altcat_1.html#G2">AltCatStr</a>(# <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">M</font>,<font color="Maroon">c</font> #) <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K6">FuncComp</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">i</font>,<font color="Maroon">j</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">j</font>,<font color="Maroon">k</font>)
 <b>by </b><i><a class="txt" href="orders_3.html#E2:36_1"><i><font color="Green">E21</font></i></a></i>;<br/>


</div>
