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

<b>let </b><font color="Maroon">C</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="altcat_1.html#L2" target="_self">AltCatStr</a> ;<br/>

<b>assume </b><a NAME="E1:48_1_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">C</font> is <a href="altcat_1.html#V8" target="_self">quasi-functional</a> &amp; <font color="Maroon">C</font> is <a href="altcat_1.html#V9" target="_self">semi-functional</a> &amp; <font color="Maroon">C</font> is <a href="altcat_1.html#V2" target="_self">transitive</a> )
 ;<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>, <font color="Maroon">o3</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D15" target="_self">ALTCAT_1:def 15</a><br/>

<b>set </b><font color="Maroon">c</font> = the <a href="altcat_1.html#U2" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>,<font color="Maroon">o3</font>;<br/>
<b>set </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><a href="altcat_1.html#K6" target="_self">FuncComp</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o2</font>,<font color="Maroon">o3</font></span>)</span></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"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<a NAME="E2:48_1_1"/>
(  not <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:48_1_1"/><b>then </b><i><font color="Green">E27</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <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"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></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><a class="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>;<br/>
<a NAME="E4:48_1_1"/><i><font color="Green">E28</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>,<font color="Maroon">o3</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_1_1"/>
( <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1" target="_self">^&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" target="_self">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_1_1"/>
( <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:48_1_1_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></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><a class="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>;<br/><b>hence </b>the <a href="altcat_1.html#U2" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>,<font color="Maroon">o3</font> = 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="altcat_1.html#K6" target="_self">FuncComp</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o2</font>,<font color="Maroon">o3</font></span>)</span></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"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i>, <a class="ref" href="relat_1.html#T110">RELAT_1:110</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_1_2"/><i><font color="Green">E36</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1" target="_self">^&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" target="_self">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:48_1_1_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E3:48_1_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K6" target="_self">FuncComp</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o2</font>,<font color="Maroon">o3</font></span>)</span></span>)</span> <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"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o2</font>,<font color="Maroon">o3</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E4:48_1_1_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_1.html#K6" target="_self">FuncComp</a> <span class="p3">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span>,<span class="p3">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o2</font>,<font color="Maroon">o3</font></span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>,<span class="p3"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <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"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o2</font>,<font color="Maroon">o3</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E5:48_1_1_1_2"/><i><font color="Green">E91</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o2</font>,<font color="Maroon">o3</font> &amp; <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E6:48_1_1_1_2"/><b>then </b><i><font color="Green">E92</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></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"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o2</font>,<font color="Maroon">o3</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</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/><a NAME="E7:48_1_1_1_2"/><i><font color="Green">E93</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>,<font color="Maroon">o3</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="altcat_1.html#E1:48_1_1_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E8:48_1_1_1_2"/><b>then </b><i><font color="Green">E101</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>,<font color="Maroon">o3</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_1.html#K6" target="_self">FuncComp</a> <span class="p3">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span>,<span class="p3">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o2</font>,<font color="Maroon">o3</font></span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>,<span class="p3"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:48_1_1_1_2_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>,<font color="Maroon">o3</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">i1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">i2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:48_1_1_1_2_1"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span> )
 <b>and </b><br/><a NAME="E4:48_1_1_1_2_1"/><i><font color="Green">E104</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">i1</font>,<font color="Maroon">i2</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">g</font> = <font color="Maroon">i1</font>, <font color="Maroon">h</font> = <font color="Maroon">i2</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="altcat_1.html#T8" target="_self">Th8</a>, <a class="ref" href="altcat_1.html#T11" target="_self">Th11</a>, <a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/><b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">i1</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o2</font>,<font color="Maroon">o3</font> <b>by </b><i><a class="ref" href="altcat_1.html#T11" target="_self">Th11</a></i>;<br/><b>reconsider </b><font color="Maroon">a2</font> = <font color="Maroon">i2</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o2</font> <b>by </b><i><a class="ref" href="altcat_1.html#T11" target="_self">Th11</a></i>;<br/><b>thus </b><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>,<font color="Maroon">o3</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>,<font color="Maroon">o3</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font>
<b>by </b><i><a class="ref" href="altcat_1.html#T12" target="_self">Th12</a></i>
<br/>.= 
<font color="Maroon">a1</font> <a href="altcat_1.html#K5" target="_self">*</a> <font color="Maroon">a2</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h</font>
<b>by </b><i>, , <a class="txt" href="altcat_1.html#E1:48_1_1_1_2"><i><font color="Green">E36</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="altcat_1.html#K6" target="_self">FuncComp</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o2</font>,<font color="Maroon">o3</font></span>)</span></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">g</font>,<font color="Maroon">h</font>
<b>by </b><i><a class="ref" href="altcat_1.html#T8" target="_self">Th8</a>, <a class="ref" href="altcat_1.html#T11" target="_self">Th11</a>, <a class="ref" href="altcat_1.html#T2" target="_self">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_1.html#K6" target="_self">FuncComp</a> <span class="p3">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span>,<span class="p3">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o2</font>,<font color="Maroon">o3</font></span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>,<span class="p3"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="altcat_1.html#T9" target="_self">Th9</a>, <a class="ref" href="altcat_1.html#T10" target="_self">Th10</a>, <a class="ref" href="altcat_1.html#T12" target="_self">Th12</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:48_1_1_1_2"/>
the <a href="altcat_1.html#U2" target="_self">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4" target="_self">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>,<font color="Maroon">o3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K6" target="_self">FuncComp</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">o2</font>,<font color="Maroon">o3</font></span>)</span></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"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1" target="_self">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1" target="_self">^&gt;</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="altcat_1.html#T9" target="_self">Th9</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
