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

<b>let </b><font color="Maroon">C</font> be   <a href="altcat_1.html#NM4">category</a>;<br/>

<a NAME="E1:56"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" target="_self">AllIso</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:56"/><b>then </b><i><font color="Green">E24</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" target="_self">AllIso</a> <font color="Maroon">C</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K3" target="_self">AllRetr</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:56"/>
 <a href="altcat_4.html#K5" target="_self">AllIso</a> <font color="Maroon">C</font> is    <a href="altcat_2.html#M1">SubCatStr</a> of  <a href="altcat_4.html#K3" target="_self">AllRetr</a> <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>proof </b></a><div class="add">

<a NAME="E1:56_1"/>
the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" target="_self">AllIso</a> <font color="Maroon">C</font></span>)</span> <a href="altcat_2.html#R2">cc=</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K3" target="_self">AllRetr</a> <font color="Maroon">C</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:56_1_1"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="altcat_4.html#K5" target="_self">AllIso</a> <font color="Maroon">C</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="altcat_4.html#K5" target="_self">AllIso</a> <font color="Maroon">C</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">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="altcat_4.html#K3" target="_self">AllRetr</a> <font color="Maroon">C</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="altcat_4.html#K3" target="_self">AllRetr</a> <font color="Maroon">C</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>; <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="E2:56_1_1"/><i><font color="Green">E25</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 <span class="p2">(<span class="default"><a href="altcat_4.html#K5" target="_self">AllIso</a> <font color="Maroon">C</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="altcat_4.html#K5" target="_self">AllIso</a> <font color="Maroon">C</font></span>)</span></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="E4:56_1_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">o1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" target="_self">AllIso</a> <font color="Maroon">C</font></span>)</span> &amp; <font color="Maroon">o2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" target="_self">AllIso</a> <font color="Maroon">C</font></span>)</span> &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><a class="txt" href="altcat_4.html#E7:56_1_1"><i><font color="Green">E37</font></i></a>, </i>;<br/>
<b>let </b><font color="Maroon">m</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="E6:56_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" target="_self">AllIso</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E7:56_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" target="_self">AllIso</a> <font color="Maroon">C</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 
<b>by </b><i><a class="txt" href="altcat_4.html#E7:56_1_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E8:56_1_1"/>
the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" target="_self">AllIso</a> <font color="Maroon">C</font></span>)</span> <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/>
<a NAME="E9:56_1_1"/><b>then </b>
the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" target="_self">AllIso</a> <font color="Maroon">C</font></span>)</span> <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> <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">o1</font>,<font color="Maroon">o2</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, <a class="txt" href="altcat_4.html#E7:56_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="altcat_2.html#D2">ALTCAT_2:def 2</a></i>;<br/>
<a NAME="E10:56_1_1"/><b>then </b>
the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" target="_self">AllIso</a> <font color="Maroon">C</font></span>)</span> <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> <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">o1</font>,<font color="Maroon">o2</font>
 
;<br/>
<a NAME="E11:56_1_1"/><b>then </b>
the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" target="_self">AllIso</a> <font color="Maroon">C</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</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">o1</font>,<font color="Maroon">o2</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">m</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/>;<br/>
<a NAME="E13:56_1_1"/>
<font color="Maroon">m1</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" target="_self">AllIso</a> <font color="Maroon">C</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 
<b>by </b><i><a class="txt" href="altcat_4.html#E7:56_1_1"><i><font color="Green">E37</font></i></a>, </i>;<br/>
<a NAME="E14:56_1_1"/><b>then </b><i><font color="Green">E45</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; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o1</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/>;<br/>
<a NAME="E15:56_1_1"/>
<font color="Maroon">m1</font> is <a href="altcat_3.html#V3">iso</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E16:56_1_1"/><b>then </b>
<font color="Maroon">m1</font> is <a href="altcat_3.html#V1">retraction</a>
 
<b>by </b><i><a class="ref" href="altcat_3.html#T5">ALTCAT_3:5</a></i>;<br/>
<a NAME="E17:56_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 <span class="p1">(<span class="default"><a href="altcat_4.html#K3" target="_self">AllRetr</a> <font color="Maroon">C</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E18:56_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K3" target="_self">AllRetr</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="altcat_4.html#E7:56_1_1"><i><font color="Green">E37</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:56_1"/>
 <a href="altcat_4.html#K5" target="_self">AllIso</a> <font color="Maroon">C</font> is    <a href="altcat_2.html#M1">SubCatStr</a> of  <a href="altcat_4.html#K3" target="_self">AllRetr</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="altcat_2.html#T25">ALTCAT_2:25</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> =  <a href="altcat_4.html#K5" target="_self">AllIso</a> <font color="Maroon">C</font> as  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V12">with_units</a>  <a href="altcat_2.html#M1">SubCatStr</a> of  <a href="altcat_4.html#K3" target="_self">AllRetr</a> <font color="Maroon">C</font> ;<br/>
<a NAME="E5:56"/>
<font color="Maroon">A</font> is <a href="altcat_2.html#V3">id-inheriting</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">o</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">o1</font> be   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K3" target="_self">AllRetr</a> <font color="Maroon">C</font></span>)</span>;<br/><b>assume </b><a NAME="E1:56_2_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font>
 ;<br/><b>reconsider </b><font color="Maroon">oo</font> = <font color="Maroon">o</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font> <b>by </b><i/>;<br/> <a href="altcat_1.html#K8">idm</a> <font color="Maroon">o</font> = 
 <a href="altcat_1.html#K8">idm</a> <font color="Maroon">oo</font>
<b>by </b><i><a class="ref" href="altcat_2.html#T35">ALTCAT_2:35</a></i>
<br/>.= 
 <a href="altcat_1.html#K8">idm</a> <font color="Maroon">o1</font>
<b>by </b><i>, <a class="ref" href="altcat_2.html#T35">ALTCAT_2:35</a></i>
;<br/><b>hence </b><a NAME="E4:56_2_1"/>
 <a href="altcat_1.html#K8">idm</a> <font color="Maroon">o1</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">o</font>,<font color="Maroon">o</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:56_2"/>
<font color="Maroon">A</font> is <a href="altcat_2.html#V3">id-inheriting</a>
 <b>by </b><i><a class="ref" href="altcat_2.html#D14">ALTCAT_2:def 14</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:56"/>
 <a href="altcat_4.html#K5" target="_self">AllIso</a> <font color="Maroon">C</font> is  non <a href="struct_0.html#V3">empty</a> <a href="altcat_2.html#NM2">subcategory</a> of  <a href="altcat_4.html#K3" target="_self">AllRetr</a> <font color="Maroon">C</font>
 ;<br/>


</div>
