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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">f</font> being <a href="functor0.html#V1">Covariant</a> <a href="functor0.html#NM1">bifunction</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> ex <font color="Olive">m</font> being   <a href="functor0.html#M1">MSUnTrans</a> of <font color="Olive">f</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">f</font>,<font color="Olive">m</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="functor0.html#G2">FunctorStr</a>(# <font color="Olive">f</font>,<font color="Olive">m</font> #) &amp; <font color="Maroon">a<sub>2</sub></font> is  <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> );<br/>
<a NAME="E1:26_1_1"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">z</font>] holds <br/><font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>given </b><font color="Maroon">f</font> being  <a href="functor0.html#V1">Covariant</a> <a href="functor0.html#NM1">bifunction</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>, <font color="Maroon">m</font> being    <a href="functor0.html#M1">MSUnTrans</a> of <font color="Maroon">f</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font><b> such that </b><a NAME="E1:26_1_1_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="functor0.html#G2">FunctorStr</a>(# <font color="Maroon">f</font>,<font color="Maroon">m</font> #) &amp; <font color="Maroon">y</font> is  <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> )
 ;<br/>

<b>given </b><font color="Maroon">f1</font> being  <a href="functor0.html#V1">Covariant</a> <a href="functor0.html#NM1">bifunction</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>, <font color="Maroon">m1</font> being    <a href="functor0.html#M1">MSUnTrans</a> of <font color="Maroon">f1</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font><b> such that </b><a NAME="E2:26_1_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">m1</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="functor0.html#G2">FunctorStr</a>(# <font color="Maroon">f1</font>,<font color="Maroon">m1</font> #) &amp; <font color="Maroon">z</font> is  <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> )
 ;<br/>

<a NAME="E3:26_1_1_1"/>
( <font color="Maroon">f</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">m</font> <a href="pboole.html#R6">=</a> <font color="Maroon">m1</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E4:26_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">A'</font> =  <a href="fraenkel.html#K1">Funcs</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>set </b><font color="Maroon">sAA</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> <a href="pboole.html#K7">*</a> <font color="Olive">f</font></span>)</span> where <font color="Olive">f</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span> : verum </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">f</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<a NAME="E3:26_1_1"/>
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> <a href="pboole.html#K7">*</a> <font color="Olive">f</font></span>)</span> where <font color="Olive">f</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span> : verum </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">sAA</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> <a href="pboole.html#K7">*</a> <font color="Olive">f</font></span>)</span> where <font color="Olive">f</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span> : verum </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">AA</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">AA</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="functor2.html#K6" target="_self">Funcs</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font>,<font color="Olive">AA</font> );<br/>
<a NAME="E5:26_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">z</font>] holds <br/><font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font>
 
;<br/>
<b>consider </b><font color="Maroon">XX</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:26_1_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font> iff  ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sAA</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="tarski.html#S1">TARSKI:sch 1</a>();<br/></i>
<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:26_1_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff  ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">XX</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="tarski.html#S1">TARSKI:sch 1</a>(<a class="txt" href="functor2.html#E1:26_1_1"><i><font color="Green">E20</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">X</font>
;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E10:26_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> implies <font color="Maroon">x</font> is  <a href="functor0.html#V9">strict</a> <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:26_1_1_2"/><b>then </b>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">XX</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  ex <font color="Olive">f</font> being <a href="functor0.html#V1">Covariant</a> <a href="functor0.html#NM1">bifunction</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> ex <font color="Olive">m</font> being   <a href="functor0.html#M1">MSUnTrans</a> of <font color="Olive">f</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">f</font>,<font color="Olive">m</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="functor0.html#G2">FunctorStr</a>(# <font color="Olive">f</font>,<font color="Olive">m</font> #) &amp; <font color="Maroon">x</font> is  <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> ) )
 
<b>by </b><i><a class="ref" href="functor2.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E3:26_1_1_2"/>
<font color="Maroon">x</font> is  <a href="functor0.html#V9">strict</a> <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>
 ;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E11:26_1_1"/>
<font color="Maroon">x</font> is  <a href="functor0.html#V9">strict</a> <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">x</font> as  <a href="functor0.html#V9">strict</a> <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> ;<br/>
<b>reconsider </b><font color="Maroon">f</font> = the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font> as  <a href="functor0.html#V1">Covariant</a> <a href="functor0.html#NM1">bifunction</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> <b>by </b><i><a class="ref" href="functor0.html#D13">FUNCTOR0:def 13</a></i>;<br/>
<a NAME="E14:26_1_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">o1</font>, <font color="Olive">o2</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>  st the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Olive">o1</font>,<font color="Olive">o2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">o1</font>,<font color="Olive">o2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>, <font color="Maroon">o2</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:26_1_1_3"/><i><font color="Green">E44</font></i>: 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:26_1_1_3"/>
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
;<br/>
<b>hence </b><a NAME="E3:26_1_1_3"/>
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="functor2.html#D2" target="_self">Def2</a>, <a class="ref" href="functor0.html#D12">FUNCTOR0:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:26_1_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">o1</font>, <font color="Olive">o2</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>  st the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Olive">o1</font>,<font color="Olive">o2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Olive">o1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Olive">o2</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>, <font color="Maroon">o2</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:26_1_1_4"/><i><font color="Green">E46</font></i>: 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:26_1_1_4"/>
<font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">o1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">o2</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="ref" href="functor0.html#T23">FUNCTOR0:23</a></i>;<br/>
<b>hence </b><a NAME="E3:26_1_1_4"/>
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">o1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">o2</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="functor2.html#T4" target="_self">Th4</a>, </i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">m</font> = the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">F</font> as    <a href="functor0.html#M1">MSUnTrans</a> of <font color="Maroon">f</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> ;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span> as    <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E18:26_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">XX</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_5"><b>proof </b></a><div class="add">

<a NAME="E1:26_1_1_5"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<a NAME="E2:26_1_1_5"/><b>then </b><i><font color="Green">E70</font></i>: 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sAA</font>
 
;<br/>
<b>reconsider </b><font color="Maroon">AA</font> = the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/>
<b>set </b><font color="Maroon">I</font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<a NAME="E4:26_1_1_5"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">m</font> is    <a href="pboole.html#M3">ManySortedFunction</a> of the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font>,<font color="Maroon">AA</font>
 
<b>by </b><i><a class="ref" href="functor0.html#D5">FUNCTOR0:def 5</a></i>;<br/>
<a NAME="E5:26_1_1_5"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_5_1"><b>proof </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><b>that </b><br/><a NAME="E1:26_1_1_5_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E2:26_1_1_5_1"/><i><font color="Green">E74</font></i>: 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> <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>consider </b><font color="Maroon">o1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">o2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:26_1_1_5_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">o1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E5:26_1_1_5_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">o2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E6:26_1_1_5_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">o1</font>, <font color="Maroon">a2</font> = <font color="Maroon">o2</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font> <b>by </b><i><a class="txt" href="functor2.html#E5:26_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="functor2.html#D3" target="_self">Def3</a></i>;<br/>
<i><font color="Green">E94</font></i>: the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</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> = 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span>
<b>by </b><i><a class="txt" href="functor2.html#E7:26_1_1"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a2</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="functor0.html#T23">FUNCTOR0:23</a></i>
;<br/>
<a NAME="E9:26_1_1_5_1"/>
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</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="functor2.html#E7:26_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:26_1_1_5_1"/>
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</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="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="functor2.html#D2" target="_self">Def2</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:26_1_1_5"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><font color="Maroon">AA</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_5_2"><b>proof </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="E1:26_1_1_5_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>assume </b><a NAME="E2:26_1_1_5_2"/><i><font color="Green">E96</font></i>: 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> <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/>

<a NAME="E3:26_1_1_5_2"/>
<font color="Maroon">AA</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</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>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>
<b>hence </b><a NAME="E4:26_1_1_5_2"/>
<font color="Maroon">AA</font> <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>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:26_1_1_5"/><b>then </b>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">AA</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<a NAME="E8:26_1_1_5"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="functor2.html#K6" target="_self">Funcs</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font>,<font color="Maroon">AA</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:26_1_1_5"/>
 <a href="functor2.html#K6" target="_self">Funcs</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font>,<font color="Maroon">AA</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XX</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:26_1_1_5"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">XX</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E11:26_1_1_5"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">XX</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E19:26_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="functor2.html#D1" target="_self">Def1</a></i>;<br/>


</div>
