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

<b>thus </b><a NAME="E1:89_1_1"/>
 <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font> is <a href="functor0.html#V12" title="FUNCTOR0:attr.12">id-preserving</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font> is <a href="functor0.html#V13" title="FUNCTOR0:attr.13">comp-preserving</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">o</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D21" target="_self" title="FUNCTOR0:def.21">FUNCTOR0:def 21</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> <span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c2">o</font></span>)</span></span><br/>

<a NAME="E1:89_1_1_1"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="altcat_2.html#D7" title="ALTCAT_2:def.7">ALTCAT_2:def 7</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="functor0.html#K5" title="FUNCTOR0:func.5">Morph-Map</a> <span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">o</font></span>)</span> = 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">o</font>
<b>by </b><i><a class="ref" href="functor0.html#T31" target="_self" title="FUNCTOR0:th.31">Th31</a></i>
<br/>.= 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c2">o</font></span>)</span>
<b>by </b><i><a class="ref" href="functor0.html#T30" target="_self" title="FUNCTOR0:th.30">Th30</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>set </b><font color="Maroon" title="c2">F</font> =  <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font>;<br/>
<a NAME="E2:89_1_1"/>
 <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font> is <a href="functor0.html#V13" title="FUNCTOR0:attr.13">comp-preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">o1</font>, <font color="Maroon" title="c4">o2</font>, <font color="Maroon" title="c5">o3</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D24" target="_self" title="FUNCTOR0:def.24">FUNCTOR0:def 24</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies for <font color="Olive" title="b1">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font><br/> for <font color="Olive" title="b2">g</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font> holds  <span class="p1">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive" title="b1">f</font></span>)</span> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:89_1_1_2"/><i><font color="Green" title="E62">A1</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>and </b><br/><a NAME="E2:89_1_1_2"/><i><font color="Green" title="E63">A2</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font><br/> for <font color="Olive" title="b2">g</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font> holds  <span class="p1">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive" title="b1">f</font></span>)</span></span><br/>

<b>let </b><font color="Maroon" title="c6">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font> holds  <span class="p1">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c6">f</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c7">g</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c7">g</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c6">f</font></span>)</span></span><br/>



<a NAME="E3:89_1_1_2"/><i><font color="Green" title="E64">A3</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c5">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="functor0.html#E1:89_1_1_2"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="functor0.html#E2:89_1_1_2"><i><font color="Green" title="E63">A2</font></i></a>, <a class="ref" href="altcat_1.html#D4" title="ALTCAT_1:def.4">ALTCAT_1:def 4</a></i>;<br/>
<a NAME="E4:89_1_1_2"/><i><font color="Green" title="E65">A4</font></i>: 
( <span class="p1">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c3">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o1</font> &amp; <span class="p1">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c4">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o2</font> &amp; <span class="p1">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">o3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">o3</font> )
 
<b>by </b><i><a class="ref" href="functor0.html#T30" target="_self" title="FUNCTOR0:th.30">Th30</a></i>;<br/>
<a NAME="E5:89_1_1_2"/>
( <span class="p1">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> &amp; <span class="p1">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> )
 
<b>by </b><i><a class="txt" href="functor0.html#E1:89_1_1_2"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="functor0.html#E2:89_1_1_2"><i><font color="Green" title="E63">A2</font></i></a>, <a class="ref" href="functor0.html#T32" target="_self" title="FUNCTOR0:th.32">Th32</a></i>;<br/>
<b>hence </b><a NAME="E6:89_1_1_2"/>
<span class="p1">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c7">g</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="functor0.html#E3:89_1_1_2"><i><font color="Green" title="E64">A3</font></i></a>, <a class="txt" href="functor0.html#E4:89_1_1_2"><i><font color="Green" title="E65">A4</font></i></a>, <a class="ref" href="functor0.html#T32" target="_self" title="FUNCTOR0:th.32">Th32</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:89_1_1"/>
 <a href="functor0.html#K11" title="FUNCTOR0:func.11">id</a> <font color="Maroon" title="c1">A</font> is <a href="functor0.html#V13" title="FUNCTOR0:attr.13">comp-preserving</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
