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

<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</a>;<br/>

<a NAME="E1:59"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K3" title="ALTCAT_4:func.3">AllRetr</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="altcat_4.html#D3" target="_self" title="ALTCAT_4:def.3">Def3</a></i>;<br/>
<a NAME="E2:59"/><b>then </b><i><font color="Green" title="E38">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K3" title="ALTCAT_4:func.3">AllRetr</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K2" title="ALTCAT_4:func.2">AllEpi</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="altcat_4.html#D2" target="_self" title="ALTCAT_4:def.2">Def2</a></i>;<br/>
<a NAME="E3:59"/>
 <a href="altcat_4.html#K3" title="ALTCAT_4:func.3">AllRetr</a> <font color="Maroon" title="c1">C</font> is    <a href="altcat_2.html#M1" title="ALTCAT_2:mode.1">SubCatStr</a> of  <a href="altcat_4.html#K2" title="ALTCAT_4:func.2">AllEpi</a> <font color="Maroon" title="c1">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<a NAME="E1:59_1"/>
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K3" title="ALTCAT_4:func.3">AllRetr</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="altcat_2.html#R2" title="ALTCAT_2:pred.2">cc=</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K2" title="ALTCAT_4:func.2">AllEpi</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:59_1_1"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="altcat_4.html#K3" title="ALTCAT_4:func.3">AllRetr</a> <font color="Maroon" title="c1">C</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="altcat_4.html#K3" title="ALTCAT_4:func.3">AllRetr</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="altcat_4.html#K2" title="ALTCAT_4:func.2">AllEpi</a> <font color="Maroon" title="c1">C</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="altcat_4.html#K2" title="ALTCAT_4:func.2">AllEpi</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="altcat_4.html#E2:59"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_2.html#D2" title="ALTCAT_2:def.2">ALTCAT_2:def 2</a><br/>

<b>let </b><font color="Maroon" title="c2">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E2:59_1_1"/><i><font color="Green" title="E39">A2</font></i>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="altcat_4.html#K3" title="ALTCAT_4:func.3">AllRetr</a> <font color="Maroon" title="c1">C</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="altcat_4.html#K3" title="ALTCAT_4:func.3">AllRetr</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">o1</font>, <font color="Maroon" title="c4">o2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:59_1_1"/><i><font color="Green" title="E40">A3</font></i>: 
( <font color="Maroon" title="c3">o1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K3" title="ALTCAT_4:func.3">AllRetr</a> <font color="Maroon" title="c1">C</font></span>)</span> &amp; <font color="Maroon" title="c4">o2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K3" title="ALTCAT_4:func.3">AllRetr</a> <font color="Maroon" title="c1">C</font></span>)</span> &amp; <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">o1</font> = <font color="Maroon" title="c3">o1</font>, <font color="Maroon" title="c6">o2</font> = <font color="Maroon" title="c4">o2</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="txt" href="altcat_4.html#E4:59_1_1"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="altcat_4.html#D3" target="_self" title="ALTCAT_4:def.3">Def3</a></i>;<br/>
<b>let </b><font color="Maroon" title="c7">m</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E6:59_1_1"/><i><font color="Green" title="E41">A4</font></i>: 
<font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K3" title="ALTCAT_4:func.3">AllRetr</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font>
 ;<br/>

<a NAME="E7:59_1_1"/><b>then </b><i><font color="Green" title="E42">A5</font></i>: 
<font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K3" title="ALTCAT_4:func.3">AllRetr</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">o1</font>,<font color="Maroon" title="c6">o2</font>
 
<b>by </b><i><a class="txt" href="altcat_4.html#E4:59_1_1"><i><font color="Green" title="E40">A3</font></i></a></i>;<br/>
<a NAME="E8:59_1_1"/>
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K3" title="ALTCAT_4:func.3">AllRetr</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="altcat_2.html#R2" title="ALTCAT_2:pred.2">cc=</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="altcat_4.html#D3" target="_self" title="ALTCAT_4:def.3">Def3</a></i>;<br/>
<a NAME="E9:59_1_1"/><b>then </b>
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K3" title="ALTCAT_4:func.3">AllRetr</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">o1</font>,<font color="Maroon" title="c6">o2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">o1</font>,<font color="Maroon" title="c6">o2</font>
 
<b>by </b><i><a class="txt" href="altcat_4.html#E2:59_1_1"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="altcat_4.html#E4:59_1_1"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="altcat_2.html#D2" title="ALTCAT_2:def.2">ALTCAT_2:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">m1</font> = <font color="Maroon" title="c7">m</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">o1</font>,<font color="Maroon" title="c6">o2</font> <b>by </b><i><a class="txt" href="altcat_4.html#E4:59_1_1"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="altcat_4.html#E6:59_1_1"><i><font color="Green" title="E41">A4</font></i></a></i>;<br/>
<a NAME="E11:59_1_1"/>
<font color="Maroon" title="c8">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K3" title="ALTCAT_4:func.3">AllRetr</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">o1</font>,<font color="Maroon" title="c6">o2</font>
 
<b>by </b><i><a class="txt" href="altcat_4.html#E4:59_1_1"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="altcat_4.html#E6:59_1_1"><i><font color="Green" title="E41">A4</font></i></a></i>;<br/>
<a NAME="E12:59_1_1"/><b>then </b><i><font color="Green" title="E43">A6</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="c5">o1</font>,<font color="Maroon" title="c6">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="c6">o2</font>,<font color="Maroon" title="c5">o1</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_4.html#D3" target="_self" title="ALTCAT_4:def.3">Def3</a></i>;<br/>
<a NAME="E13:59_1_1"/>
<font color="Maroon" title="c8">m1</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a>
 
<b>by </b><i><a class="txt" href="altcat_4.html#E7:59_1_1"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="altcat_4.html#D3" target="_self" title="ALTCAT_4:def.3">Def3</a></i>;<br/>
<a NAME="E14:59_1_1"/><b>then </b>
<font color="Maroon" title="c8">m1</font> is <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a>
 
<b>by </b><i><a class="txt" href="altcat_4.html#E12:59_1_1"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="altcat_3.html#T15" title="ALTCAT_3:th.15">ALTCAT_3:15</a></i>;<br/>
<a NAME="E15:59_1_1"/><b>then </b>
<font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K2" title="ALTCAT_4:func.2">AllEpi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">o1</font>,<font color="Maroon" title="c6">o2</font>
 
<b>by </b><i><a class="txt" href="altcat_4.html#E12:59_1_1"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="altcat_4.html#D2" target="_self" title="ALTCAT_4:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E16:59_1_1"/>
<font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K2" title="ALTCAT_4:func.2">AllEpi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font>
 <b>by </b><i><a class="txt" href="altcat_4.html#E4:59_1_1"><i><font color="Green" title="E40">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:59_1"/>
 <a href="altcat_4.html#K3" title="ALTCAT_4:func.3">AllRetr</a> <font color="Maroon" title="c1">C</font> is    <a href="altcat_2.html#M1" title="ALTCAT_2:mode.1">SubCatStr</a> of  <a href="altcat_4.html#K2" title="ALTCAT_4:func.2">AllEpi</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="altcat_4.html#E2:59"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="altcat_2.html#T25" title="ALTCAT_2:th.25">ALTCAT_2:25</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c2">A</font> =  <a href="altcat_4.html#K3" title="ALTCAT_4:func.3">AllRetr</a> <font color="Maroon" title="c1">C</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="altcat_1.html#V12" title="ALTCAT_1:attr.12">with_units</a>  <a href="altcat_2.html#M1" title="ALTCAT_2:mode.1">SubCatStr</a> of  <a href="altcat_4.html#K2" title="ALTCAT_4:func.2">AllEpi</a> <font color="Maroon" title="c1">C</font> ;<br/>
<a NAME="E5:59"/>
<font color="Maroon" title="c2">A</font> is <a href="altcat_2.html#V3" title="ALTCAT_2:attr.3">id-inheriting</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">o</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">A</font>;<br/><b>let </b><font color="Maroon" title="c4">o1</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K2" title="ALTCAT_4:func.2">AllEpi</a> <font color="Maroon" title="c1">C</font></span>)</span>;<br/><b>assume </b><a NAME="E1:59_2_1"/><i><font color="Green" title="E39">A7</font></i>: 
<font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o1</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c5">oo</font> = <font color="Maroon" title="c3">o</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="ref" href="altcat_4.html#D3" target="_self" title="ALTCAT_4:def.3">Def3</a></i>;<br/> <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">o</font> = 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c5">oo</font>
<b>by </b><i><a class="ref" href="altcat_2.html#T35" title="ALTCAT_2:th.35">ALTCAT_2:35</a></i>
<br/>.= 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c4">o1</font>
<b>by </b><i><a class="txt" href="altcat_4.html#E1:59_2_1"><i><font color="Green" title="E39">A7</font></i></a>, <a class="ref" href="altcat_2.html#T35" title="ALTCAT_2:th.35">ALTCAT_2:35</a></i>
;<br/><b>hence </b><a NAME="E4:59_2_1"/>
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c4">o1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c3">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:59_2"/>
<font color="Maroon" title="c2">A</font> is <a href="altcat_2.html#V3" title="ALTCAT_2:attr.3">id-inheriting</a>
 <b>by </b><i><a class="ref" href="altcat_2.html#D14" title="ALTCAT_2:def.14">ALTCAT_2:def 14</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:59"/>
 <a href="altcat_4.html#K3" title="ALTCAT_4:func.3">AllRetr</a> <font color="Maroon" title="c1">C</font> is  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="altcat_2.html#NM2" title="ALTCAT_2:NM.2">subcategory</a> of  <a href="altcat_4.html#K2" title="ALTCAT_4:func.2">AllEpi</a> <font color="Maroon" title="c1">C</font>
 ;<br/>


</div>
