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

<b>set </b><font color="Maroon">I</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>;<br/>
<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>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">a<sub>2</sub></font> iff  ex <font color="Olive">o1</font>, <font color="Olive">o2</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font> ex <font color="Olive">m</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">o1</font>,<font color="Olive">o2</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">o1</font>,<font color="Olive">o2</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">o1</font>,<font color="Olive">o2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> is <a href="altcat_3.html#V5">epi</a> ) );<br/>
<a NAME="E1:48_1_1"/><i><font color="Green">E24</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">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/> ex <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">X</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_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> ;<br/>

<b>assume </b><a NAME="E1:48_1_1_1"/>
<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">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>then 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="E3:48_1_1_1"/><i><font color="Green">E25</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">C</font> &amp; <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">C</font> &amp; <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#T103">ZFMISC_1:103</a></i>;<br/>
<b>reconsider </b><font color="Maroon">o1</font> = <font color="Maroon">o1</font>, <font color="Maroon">o2</font> = <font color="Maroon">o2</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font> <b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">m</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o2</font> st <br/>( <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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">m</font> is <a href="altcat_3.html#V5">epi</a> );<br/>
<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:48_1_1_1"/><i><font color="Green">E26</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 ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</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="E7:48_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> implies  ex <font color="Olive">o1</font>, <font color="Olive">o2</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font> ex <font color="Olive">m</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">o1</font>,<font color="Olive">o2</font> st <br/>( <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="Olive">o1</font>,<font color="Olive">o2</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">o1</font>,<font color="Olive">o2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> is <a href="altcat_3.html#V5">epi</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">m</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o2</font><b> such that </b><br/><a NAME="E3:48_1_1_1_1"/><i><font color="Green">E32</font></i>: 
( <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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">m</font> is <a href="altcat_3.html#V5">epi</a> )
 <b>by </b><i><a class="txt" href="altcat_4.html#E3:48_1_1_1_1"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">o1</font>
;<br/>

<b>take </b>
<font color="Maroon">o2</font>
;<br/>

<b>take </b>
<font color="Maroon">m</font>
;<br/>

<b>thus </b><a NAME="E4:48_1_1_1_1"/>
( <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> &amp; <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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> is <a href="altcat_3.html#V5">epi</a> )
 <b>by </b><i>, </i>;<br/>


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

<b>given </b><font color="Maroon">p1</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">p2</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">m</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">p1</font>,<font color="Maroon">p2</font><b> such that </b><a NAME="E8:48_1_1_1"/><i><font color="Green">E37</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">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> is <a href="altcat_3.html#V5">epi</a> )
 ;<br/>

<a NAME="E9:48_1_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>thus </b><a NAME="E10:48_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">Ar</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">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font></span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E3:48_1_1"/><i><font color="Green">E46</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">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Maroon">Ar</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>(<a class="ref" href="altcat_4.html#T4" target="_self">Th4</a>);<br/></i>
<a NAME="E4:48_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">Ar</font> <a href="altcat_2.html#R2">cc=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:48_1_1_2"/>
<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">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</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">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_2.html#D2">ALTCAT_2:def 2</a><br/>

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

<b>assume </b><a NAME="E2:48_1_1_2"/><i><font color="Green">E50</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">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>let </b><font color="Maroon">q</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>assume </b><a NAME="E3:48_1_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ar</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E4:48_1_1_2"/><b>then </b>
 ex <font color="Olive">p1</font>, <font color="Olive">p2</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font> ex <font color="Olive">m</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">p1</font>,<font color="Olive">p2</font> st <br/>( <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="Olive">p1</font>,<font color="Olive">p2</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">p1</font>,<font color="Olive">p2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> is <a href="altcat_3.html#V5">epi</a> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:48_1_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>


</div><b>end;</b></div>
<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">p1</font>, <font color="Olive">p2</font>, <font color="Olive">p3</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Olive">p1</font>,<font color="Olive">p2</font>,<font color="Olive">p3</font></span><a href="mcart_1.html#K3">]</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4">.</a> <font color="Olive">p1</font>,<font color="Olive">p2</font>,<font color="Olive">p3</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Olive">p2</font>,<font color="Olive">p3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Olive">p1</font>,<font color="Olive">p2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> );<br/>
<a NAME="E5:48_1_1"/><i><font color="Green">E52</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#K3">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font></span><a href="zfmisc_1.html#K3">:]</a></span> holds <br/> ex <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>,<font color="Olive">j</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_3"><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="E1:48_1_1_3"/>
<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">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font></span><a href="zfmisc_1.html#K3">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">p1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">p2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">p3</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:48_1_1_3"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font> &amp; <font color="Maroon">p3</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font> &amp; <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">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</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/>
<b>reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">p1</font>, <font color="Maroon">p2</font> = <font color="Maroon">p2</font>, <font color="Maroon">p3</font> = <font color="Maroon">p3</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font> <b>by </b><i/>;<br/>
<b>take </b>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
;<br/>

<b>take </b>
<font color="Maroon">p1</font>
;<br/>

<b>take </b>
<font color="Maroon">p2</font>
;<br/>

<b>take </b>
<font color="Maroon">p3</font>
;<br/>

<b>thus </b><a NAME="E5:48_1_1_3"/>
<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">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font></span><a href="mcart_1.html#K3">]</a></span>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E6:48_1_1_3"/>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">Co</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font></span><a href="zfmisc_1.html#K3">:]</a></span><b> such that </b><br/><a NAME="E7:48_1_1"/><i><font color="Green">E112</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#K3">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font></span><a href="zfmisc_1.html#K3">:]</a></span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>,<font color="Maroon">Co</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>();<br/></i>
<a NAME="E8:48_1_1"/>
<font color="Maroon">Co</font> is    <a href="pboole.html#M3">ManySortedFunction</a> of <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">Ar</font>,<font color="Maroon">Ar</font></span><a href="altcat_1.html#K3">|}</a></span>,<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">Ar</font></span><a href="altcat_1.html#K2">|}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_4"><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="pboole.html#D18">PBOOLE:def 18</a><br/>

<b>assume </b><a NAME="E1:48_1_1_4"/>
<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">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font></span><a href="zfmisc_1.html#K3">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">p1</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">p2</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">p3</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:48_1_1_4"/><i><font color="Green">E113</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">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font></span><a href="mcart_1.html#K3">]</a></span>
 <b>and </b><br/><a NAME="E4:48_1_1_4"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">Co</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i/>;<br/>
<a NAME="E5:48_1_1_4"/><i><font color="Green">E115</font></i>: 
( the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p3</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E6:48_1_1_4"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">p3</font></span><a href="tarski.html#K4">]</a></span> <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">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K4">]</a></span> <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">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font></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/>
<a NAME="E7:48_1_1_4"/><b>then </b>
( <font color="Maroon">Ar</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">p3</font></span><a href="tarski.html#K4">]</a></span> <a href="tarski.html#R1">c=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">p3</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">Ar</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K4">]</a></span> <a href="tarski.html#R1">c=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i>, <a class="ref" href="altcat_2.html#D2">ALTCAT_2:def 2</a></i>;<br/>
<a NAME="E8:48_1_1_4"/><b>then </b>
( <font color="Maroon">Ar</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">p3</font></span><a href="tarski.html#K4">]</a></span> <a href="tarski.html#R1">c=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font> &amp; <font color="Maroon">Ar</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K4">]</a></span> <a href="tarski.html#R1">c=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 
;<br/>
<a NAME="E9:48_1_1_4"/><b>then </b><i><font color="Green">E116</font></i>: 
( <font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font> <a href="tarski.html#R1">c=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font> &amp; <font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="tarski.html#R1">c=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 
;<br/>
<a NAME="E10:48_1_1_4"/><i><font color="Green">E117</font></i>: 
( the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p3</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_1_1_4_1"/><i><font color="Green">E118</font></i>: 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p3</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>assume </b><a NAME="E2:48_1_1_4_1"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <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">k</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:48_1_1_4_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">k</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"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></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">u1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">u2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:48_1_1_4_1"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font> &amp; <font color="Maroon">u2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">u2</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/>
<a NAME="E7:48_1_1_4_1"/>
( <font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font> &amp; <font color="Maroon">u2</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:48_1_1_4_1"/><b>then </b>
( <font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">p3</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Maroon">u2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 
;<br/>
<a NAME="E9:48_1_1_4_1"/><b>then </b>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p3</font></span><a href="altcat_1.html#K1">^&gt;</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="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E10:48_1_1_4_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E11:48_1_1_4"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">Co</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p3</font> <b>by </b><i>, , <a class="ref" href="funct_2.html#T38">FUNCT_2:38</a></i>;<br/>
<a NAME="E13:48_1_1_4"/><i><font color="Green">E121</font></i>: 
<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">Ar</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p3</font>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D5">ALTCAT_1:def 5</a></i>;<br/>
<a NAME="E14:48_1_1_4"/><i><font color="Green">E122</font></i>: 
<span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">Ar</font>,<font color="Maroon">Ar</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D6">ALTCAT_1:def 6</a></i>;<br/>
<a NAME="E15:48_1_1_4"/><i><font color="Green">E123</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">Ar</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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>assume </b><a NAME="E1:48_1_1_4_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:48_1_1_4_2"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:48_1_1_4_2"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:48_1_1_4_2"/><i><font color="Green">E126</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">m1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">m2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:48_1_1_4_2"/><i><font color="Green">E128</font></i>: 
( <font color="Maroon">m1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font> &amp; <font color="Maroon">m2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 <b>and </b><br/><a NAME="E8:48_1_1_4_2"/><i><font color="Green">E129</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">m1</font>,<font color="Maroon">m2</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/>
<a NAME="E9:48_1_1_4_2"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">m1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ar</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">p3</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">m2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ar</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i><a class="txt" href="altcat_4.html#E3:48_1_1"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E10:48_1_1_4_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">p3</font></span><a href="tarski.html#K4">]</a></span> <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">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font></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/>
<b>then consider </b><font color="Maroon">q2</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">q3</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">qq</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">q2</font>,<font color="Maroon">q3</font><b> such that </b><br/><a NAME="E12:48_1_1_4_2"/><i><font color="Green">E134</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">p3</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">q2</font>,<font color="Maroon">q3</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">q2</font>,<font color="Maroon">q3</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">qq</font> &amp; <font color="Maroon">qq</font> is <a href="altcat_3.html#V5">epi</a> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E13:48_1_1_4_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K4">]</a></span> <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">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font></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/>
<b>then consider </b><font color="Maroon">r1</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">r2</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">rr</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">r1</font>,<font color="Maroon">r2</font><b> such that </b><br/><a NAME="E15:48_1_1_4_2"/><i><font color="Green">E138</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">m2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rr</font> &amp; <font color="Maroon">rr</font> is <a href="altcat_3.html#V5">epi</a> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E16:48_1_1_4_2"/><i><font color="Green">E139</font></i>: 
 ex <font color="Olive">o1</font>, <font color="Olive">o3</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font> ex <font color="Olive">m</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">o1</font>,<font color="Olive">o3</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p3</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o1</font>,<font color="Olive">o3</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">o1</font>,<font color="Olive">o3</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> is <a href="altcat_3.html#V5">epi</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_4_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:48_1_1_4_2_1"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E2:48_1_1_4_2_1"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">mm</font> = <font color="Maroon">qq</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">r2</font>,<font color="Maroon">q3</font> <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>take </b>
<font color="Maroon">r1</font>
;<br/>

<b>take </b>
<font color="Maroon">q3</font>
;<br/>

<b>take </b>
<font color="Maroon">mm</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">rr</font>
;<br/>

<a NAME="E4:48_1_1_4_2_1"/><i><font color="Green">E143</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q3</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E5:48_1_1_4_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p3</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">q3</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>

<b>thus </b><a NAME="E6:48_1_1_4_2_1"/>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">q3</font></span><a href="altcat_1.html#K1">^&gt;</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="txt" href="altcat_4.html#E5:48_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="altcat_4.html#T13" target="_self">Th13</a>, <a class="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<br/>

<b>thus </b><font color="Maroon">q</font> = 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">mm</font>,<font color="Maroon">rr</font>
<b>by </b><i>, , , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a>, <a class="txt" href="altcat_4.html#E4:48_1_1"><i><font color="Green">E47</font></i></a>, , </i>
<br/>.= 
<font color="Maroon">mm</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">rr</font>
<b>by </b><i>, , <a class="txt" href="altcat_4.html#E5:48_1_1"><i><font color="Green">E52</font></i></a>, , <a class="ref" href="altcat_1.html#D10">ALTCAT_1:def 10</a></i>
;<br/>

<b>thus </b><a NAME="E8:48_1_1_4_2_1"/>
<font color="Maroon">mm</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">rr</font> is <a href="altcat_3.html#V5">epi</a>
 <b>by </b><i>, , <a class="txt" href="altcat_4.html#E5:48_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="altcat_4.html#T13" target="_self">Th13</a>, <a class="ref" href="altcat_3.html#T10">ALTCAT_3:10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:48_1_1_4_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p3</font></span><a href="tarski.html#K4">]</a></span> <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">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font></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/>
<a NAME="E18:48_1_1_4_2"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ar</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p3</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E19:48_1_1_4_2"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p3</font>
 
;<br/>
<b>hence </b><a NAME="E20:48_1_1_4_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">Ar</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="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:48_1_1_4"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">Ar</font>,<font color="Maroon">Ar</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/>
<b>hence </b><a NAME="E17:48_1_1_4"/>
<font color="Maroon">Co</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">Ar</font>,<font color="Maroon">Ar</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">Ar</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="ref" href="funct_2.html#T8">FUNCT_2:8</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Co</font> = <font color="Maroon">Co</font> as   <a href="altcat_1.html#NM3">BinComp</a> of <font color="Maroon">Ar</font> ;<br/>
<b>set </b><font color="Maroon">IT</font> =  <a href="altcat_1.html#G2">AltCatStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>,<font color="Maroon">Ar</font>,<font color="Maroon">Co</font> #);<br/>
<b>set </b><font color="Maroon">J</font> = the <a href="struct_0.html#U1">carrier</a> of <a href="altcat_1.html#G2">AltCatStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>,<font color="Maroon">Ar</font>,<font color="Maroon">Co</font> #);<br/>
<a NAME="E10:48_1_1"/>
 <a href="altcat_1.html#G2">AltCatStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>,<font color="Maroon">Ar</font>,<font color="Maroon">Co</font> #) is    <a href="altcat_2.html#M1">SubCatStr</a> of <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:48_1_1_5"/>
the <a href="struct_0.html#U1">carrier</a> of <a href="altcat_1.html#G2">AltCatStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>,<font color="Maroon">Ar</font>,<font color="Maroon">Co</font> #) <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_2.html#D11">ALTCAT_2:def 11</a><br/>

<b>thus </b><a NAME="E2:48_1_1_5"/>
the <a href="altcat_1.html#U1">Arrows</a> of <a href="altcat_1.html#G2">AltCatStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>,<font color="Maroon">Ar</font>,<font color="Maroon">Co</font> #) <a href="altcat_2.html#R2">cc=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E3:48_1_1_5"/>
<span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="altcat_1.html#G2">AltCatStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>,<font color="Maroon">Ar</font>,<font color="Maroon">Co</font> #),the <a href="struct_0.html#U1">carrier</a> of <a href="altcat_1.html#G2">AltCatStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>,<font color="Maroon">Ar</font>,<font color="Maroon">Co</font> #),the <a href="struct_0.html#U1">carrier</a> of <a href="altcat_1.html#G2">AltCatStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>,<font color="Maroon">Ar</font>,<font color="Maroon">Co</font> #)</span><a href="zfmisc_1.html#K3">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font></span><a href="zfmisc_1.html#K3">:]</a></span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_2.html#D2">ALTCAT_2:def 2</a><br/>

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

<b>assume </b><a NAME="E4:48_1_1_5"/><i><font color="Green">E146</font></i>: 
<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">the <a href="struct_0.html#U1">carrier</a> of <a href="altcat_1.html#G2">AltCatStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>,<font color="Maroon">Ar</font>,<font color="Maroon">Co</font> #),the <a href="struct_0.html#U1">carrier</a> of <a href="altcat_1.html#G2">AltCatStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>,<font color="Maroon">Ar</font>,<font color="Maroon">Co</font> #),the <a href="struct_0.html#U1">carrier</a> of <a href="altcat_1.html#G2">AltCatStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>,<font color="Maroon">Ar</font>,<font color="Maroon">Co</font> #)</span><a href="zfmisc_1.html#K3">:]</a></span>
 ;<br/>

<b>let </b><font color="Maroon">q</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>assume </b><a NAME="E5:48_1_1_5"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U2">Comp</a> of <a href="altcat_1.html#G2">AltCatStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>,<font color="Maroon">Ar</font>,<font color="Maroon">Co</font> #) <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<b>consider </b><font color="Maroon">p1</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">p2</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">p3</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E7:48_1_1_5"/><i><font color="Green">E148</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">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font></span><a href="mcart_1.html#K3">]</a></span> &amp; <font color="Maroon">Co</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:48_1_1_5"/>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T88">RELAT_1:88</a></i>;<br/>
<a NAME="E9:48_1_1_5"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E10:48_1_1_5"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C</font> <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/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">IT</font> =  <a href="altcat_1.html#G2">AltCatStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>,<font color="Maroon">Ar</font>,<font color="Maroon">Co</font> #) as  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V6">strict</a>  <a href="altcat_2.html#M1">SubCatStr</a> of <font color="Maroon">C</font> ;<br/>
<a NAME="E12:48_1_1"/>
<font color="Maroon">IT</font> is <a href="altcat_1.html#V2">transitive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">IT</font>, <font color="Maroon">p2</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">IT</font>, <font color="Maroon">p3</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">IT</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a><br/>

<b>assume </b><a NAME="E1:48_1_1_6"/><i><font color="Green">E149</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">p3</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <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">m2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:48_1_1_6"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">m2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="altcat_1.html#K1">^&gt;</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">m1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:48_1_1_6"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">m1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">p3</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E6:48_1_1_6"/>
( <font color="Maroon">m2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; <font color="Maroon">m1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:48_1_1_6"/><b>then </b><i><font color="Green">E152</font></i>: 
( <font color="Maroon">m1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ar</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">p3</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">m2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ar</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K4">]</a></span> )
 
;<br/>
<a NAME="E8:48_1_1_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">p3</font></span><a href="tarski.html#K4">]</a></span> <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">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font></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/>
<b>then consider </b><font color="Maroon">q2</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">q3</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">qq</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">q2</font>,<font color="Maroon">q3</font><b> such that </b><br/><a NAME="E10:48_1_1_6"/><i><font color="Green">E153</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">p3</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">q2</font>,<font color="Maroon">q3</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">q2</font>,<font color="Maroon">q3</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">qq</font> &amp; <font color="Maroon">qq</font> is <a href="altcat_3.html#V5">epi</a> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E11:48_1_1_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K4">]</a></span> <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">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font></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/>
<b>then consider </b><font color="Maroon">r1</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">r2</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">rr</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">r1</font>,<font color="Maroon">r2</font><b> such that </b><br/><a NAME="E13:48_1_1_6"/><i><font color="Green">E154</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">m2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rr</font> &amp; <font color="Maroon">rr</font> is <a href="altcat_3.html#V5">epi</a> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E14:48_1_1_6"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E15:48_1_1_6"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">mm</font> = <font color="Maroon">qq</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">r2</font>,<font color="Maroon">q3</font> <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E17:48_1_1_6"/><i><font color="Green">E157</font></i>: 
 ex <font color="Olive">o1</font>, <font color="Olive">o3</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font> ex <font color="Olive">m</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">o1</font>,<font color="Olive">o3</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p3</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">o1</font>,<font color="Olive">o3</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">o1</font>,<font color="Olive">o3</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">mm</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">rr</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> is <a href="altcat_3.html#V5">epi</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_6_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">r1</font>
;<br/>

<b>take </b>
<font color="Maroon">q3</font>
;<br/>

<b>take </b>
<font color="Maroon">mm</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">rr</font>
;<br/>

<a NAME="E1:48_1_1_6_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q3</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E2:48_1_1_6_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p3</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">q3</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>

<b>thus </b><a NAME="E3:48_1_1_6_1"/>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">q3</font></span><a href="altcat_1.html#K1">^&gt;</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="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<br/>

<b>thus </b><a NAME="E4:48_1_1_6_1"/>
<font color="Maroon">mm</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">rr</font> <a href="hidden.html#R1">=</a> <font color="Maroon">mm</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">rr</font>
 ;<br/>

<b>thus </b><a NAME="E5:48_1_1_6_1"/>
<font color="Maroon">mm</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">rr</font> is <a href="altcat_3.html#V5">epi</a>
 <b>by </b><i>, , , , <a class="ref" href="altcat_3.html#T10">ALTCAT_3:10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:48_1_1_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p3</font></span><a href="tarski.html#K4">]</a></span> <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">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font></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/>
<a NAME="E19:48_1_1_6"/><b>then </b>
<font color="Maroon">mm</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">rr</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ar</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p3</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E20:48_1_1_6"/><b>then </b>
<font color="Maroon">mm</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">rr</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ar</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">p1</font>,<font color="Maroon">p3</font>
 
;<br/>
<b>hence </b><a NAME="E21:48_1_1_6"/>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p3</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">IT</font> = <font color="Maroon">IT</font> as  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V2">transitive</a> <a href="altcat_1.html#V6">strict</a>  <a href="altcat_2.html#M1">SubCatStr</a> of <font color="Maroon">C</font> ;<br/>
<b>take </b>
<font color="Maroon">IT</font>
;<br/>

<b>thus </b><a NAME="E14:48_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>
 ;<br/>

<b>thus </b><a NAME="E15:48_1_1"/>
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">IT</font> <a href="altcat_2.html#R2">cc=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">o1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">o2</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">m</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o2</font>;<br/>



<a NAME="E16:48_1_1"/><i><font color="Green">E158</font></i>: 
<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> <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">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font></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/>
<b>thus </b><a NAME="E17:48_1_1"/>
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">IT</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font> implies ( <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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">m</font> is <a href="altcat_3.html#V5">epi</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_1_1_7"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">IT</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 ;<br/>

<a NAME="E2:48_1_1_7"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ar</font> <a href="funct_1.html#K1">.</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>
 
;<br/>
<b>then consider </b><font color="Maroon">p1</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">p2</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">n</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">p1</font>,<font color="Maroon">p2</font><b> such that </b><br/><a NAME="E4:48_1_1_7"/><i><font color="Green">E159</font></i>: 
( <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> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> is <a href="altcat_3.html#V5">epi</a> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E5:48_1_1_7"/>
( <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E6:48_1_1_7"/>
( <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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">m</font> is <a href="altcat_3.html#V5">epi</a> )
 <b>by </b><i/>;<br/>


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

<b>assume </b><a NAME="E18:48_1_1"/>
( <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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">m</font> is <a href="altcat_3.html#V5">epi</a> )
 ;<br/>

<a NAME="E19:48_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</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>, </i>;<br/>
<b>hence </b><a NAME="E20:48_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">IT</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 ;<br/>


</div>
