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

<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be   <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</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="functor0.html#V16" title="FUNCTOR0:attr.16">contravariant</a>  <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>  st <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <a href="yellow20.html#R3" title="YELLOW20:pred.3">are_anti-isomorphic_under</a> <font color="Olive" title="b1">F</font> holds <br/><font color="Olive" title="b1">F</font> is <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a></span><br/><b>let </b><font color="Maroon" title="c3">F</font> be  <a href="functor0.html#V16" title="FUNCTOR0:attr.16">contravariant</a>  <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <a href="yellow20.html#R3" title="YELLOW20:pred.3">are_anti-isomorphic_under</a> <font color="Maroon" title="c3">F</font> implies <font color="Maroon" title="c3">F</font> is <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a> )</span><br/>



<b>assume </b><a NAME="E1:53"/>
( <font color="Maroon" title="c1">A</font> is   <a href="altcat_2.html#NM2" title="ALTCAT_2:NM.2">subcategory</a> of <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">B</font> is   <a href="altcat_2.html#NM2" title="ALTCAT_2:NM.2">subcategory</a> of <font color="Maroon" title="c2">B</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow20.html#D5" target="_self" title="YELLOW20:def.5">YELLOW20:def 5</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">G</font> being <a href="functor0.html#V16" title="FUNCTOR0:attr.16">contravariant</a>  <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>  holds <br/>(  not <font color="Olive" title="b1">G</font> is <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a> or  ex <font color="Olive" title="b2">a'</font>, <font color="Olive" title="b3">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b2">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a</font> &amp; not <font color="Olive" title="b1">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b2">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b3">a</font> ) or  ex <font color="Olive" title="b2">b'</font>, <font color="Olive" title="b3">c'</font>, <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">c</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font> st <br/>( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b2">b'</font>,<font color="Olive" title="b3">c'</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; <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b</font> &amp; <font color="Olive" title="b3">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">c</font> &amp;  ex <font color="Olive" title="b6">f'</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b2">b'</font>,<font color="Olive" title="b3">c'</font> ex <font color="Olive" title="b7">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">c</font> st <br/>( <font color="Olive" title="b6">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">f</font> &amp; not <font color="Olive" title="b1">G</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b6">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> <font color="Maroon" title="c3">F</font>,<font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">f</font> ) ) ) or <font color="Maroon" title="c3">F</font> is <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a> )</span><br/>

<b>given </b><font color="Maroon" title="c4">G</font> being  <a href="functor0.html#V16" title="FUNCTOR0:attr.16">contravariant</a>  <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font><b> such that </b><a NAME="E3:53"/><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon" title="c4">G</font> is <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a>
 <b>and </b><br/><a NAME="E4:53"/><i><font color="Green" title="E33">A2</font></i>: 
for <font color="Olive" title="b1">a'</font>, <font color="Olive" title="b2">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> holds <br/><font color="Maroon" title="c4">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b2">a</font>
 <b>and </b><br/><a NAME="E5:53"/><i><font color="Green" title="E34">A3</font></i>: 
for <font color="Olive" title="b1">b'</font>, <font color="Olive" title="b2">c'</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>  st <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">b'</font>,<font color="Olive" title="b2">c'</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; <font color="Olive" title="b1">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b2">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">c</font> holds <br/>for <font color="Olive" title="b5">f'</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">b'</font>,<font color="Olive" title="b2">c'</font><br/> for <font color="Olive" title="b6">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font>  st <font color="Olive" title="b5">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">f</font> holds <br/><font color="Maroon" title="c4">G</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b5">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K7" title="FUNCTOR0:func.7">Morph-Map</a> <font color="Maroon" title="c3">F</font>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">F</font> is <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a></span><br/>

<a NAME="E6:53"/><i><font color="Green" title="E35">A4</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font> holds  <font color="Maroon" title="c3">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a</font>
 
<b>by </b><i><a class="txt" href="yellow20.html#E4:53"><i><font color="Green" title="E33">A2</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>; <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="c5">a</font>,<font color="Maroon" title="c6">b</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="c5">a</font>,<font color="Maroon" title="c6">b</font> holds  <font color="Maroon" title="c3">F</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b1">f</font> )</span><br/><b>assume </b><a NAME="E1:53_1"/><i><font color="Green" title="E36">A5</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">a</font>,<font color="Maroon" title="c6">b</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="c5">a</font>,<font color="Maroon" title="c6">b</font> holds  <font color="Maroon" title="c3">F</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Olive" title="b1">f</font></span><br/><b>let </b><font color="Maroon" title="c7">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">F</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c7">f</font></span><br/><a NAME="E2:53_1"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span></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="yellow20.html#E1:53_1"><i><font color="Green" title="E36">A5</font></i></a>, <a class="ref" href="functor0.html#D20" title="FUNCTOR0:def.20">FUNCTOR0:def 20</a></i>;<br/><b>hence </b><font color="Maroon" title="c3">F</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c7">f</font> = 
<span class="p1">(<span class="default"><a href="functor0.html#K7" title="FUNCTOR0:func.7">Morph-Map</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">f</font>
<b>by </b><i><a class="txt" href="yellow20.html#E1:53_1"><i><font color="Green" title="E36">A5</font></i></a>, <a class="ref" href="functor0.html#D17" title="FUNCTOR0:def.17">FUNCTOR0:def 17</a></i>
<br/>.= 
<font color="Maroon" title="c4">G</font> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c7">f</font>
<b>by </b><i><a class="txt" href="yellow20.html#E5:53"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="yellow20.html#E1:53_1"><i><font color="Green" title="E36">A5</font></i></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>
<a NAME="E8:53"/><b>then </b><i><font color="Green" title="E36">A6</font></i>: 
 <a href="functor0.html#G2" title="FUNCTOR0:aggr.2">FunctorStr</a>(# the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c3">F</font>,the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c3">F</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="functor0.html#G2" title="FUNCTOR0:aggr.2">FunctorStr</a>(# the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c4">G</font>,the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c4">G</font> #)
 
<b>by </b><i><a class="txt" href="yellow20.html#E6:53"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="yellow18.html#T2" title="YELLOW18:th.2">YELLOW18:2</a></i>;<br/>
<a NAME="E9:53"/>
( <font color="Maroon" title="c4">G</font> is <a href="functor0.html#V19" title="FUNCTOR0:attr.19">injective</a> &amp; <font color="Maroon" title="c4">G</font> is <a href="functor0.html#V20" title="FUNCTOR0:attr.20">surjective</a> )
 
<b>by </b><i><a class="txt" href="yellow20.html#E3:53"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="functor0.html#D36" title="FUNCTOR0:def.36">FUNCTOR0:def 36</a></i>;<br/>
<a NAME="E10:53"/><b>then </b>
( <font color="Maroon" title="c4">G</font> is <a href="functor0.html#V4" title="FUNCTOR0:attr.4">one-to-one</a> &amp; <font color="Maroon" title="c4">G</font> is <a href="functor0.html#V17" title="FUNCTOR0:attr.17">faithful</a> &amp; <font color="Maroon" title="c4">G</font> is <a href="functor0.html#V18" title="FUNCTOR0:attr.18">full</a> &amp; <font color="Maroon" title="c4">G</font> is <a href="functor0.html#V5" title="FUNCTOR0:attr.5">onto</a> )
 
<b>by </b><i><a class="ref" href="functor0.html#D34" title="FUNCTOR0:def.34">FUNCTOR0:def 34</a>, <a class="ref" href="functor0.html#D35" title="FUNCTOR0:def.35">FUNCTOR0:def 35</a></i>;<br/>
<b>hence </b><a NAME="E11:53"/>
( the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c3">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c3">F</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> &amp;  ex <font color="Olive" title="b1">f</font> being   <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c1">A</font>,the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c2">B</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c3">F</font> st <br/>( <font color="Olive" title="b1">f</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c3">F</font> &amp; <font color="Olive" title="b1">f</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> ) &amp; the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c3">F</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> )
 <b>by </b><i><a class="txt" href="yellow20.html#E8:53"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="functor0.html#D7" title="FUNCTOR0:def.7">FUNCTOR0:def 7</a>, <a class="ref" href="functor0.html#D8" title="FUNCTOR0:def.8">FUNCTOR0:def 8</a>, <a class="ref" href="functor0.html#D31" title="FUNCTOR0:def.31">FUNCTOR0:def 31</a>, <a class="ref" href="functor0.html#D33" title="FUNCTOR0:def.33">FUNCTOR0:def 33</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D7" title="FUNCTOR0:def.7">FUNCTOR0:def 7</a>,<a class="ref" href="functor0.html#D8" title="FUNCTOR0:def.8">FUNCTOR0:def 8</a>,<a class="ref" href="functor0.html#D31" title="FUNCTOR0:def.31">FUNCTOR0:def 31</a>,<a class="ref" href="functor0.html#D33" title="FUNCTOR0:def.33">FUNCTOR0:def 33</a>,<a class="ref" href="functor0.html#D34" title="FUNCTOR0:def.34">FUNCTOR0:def 34</a>,<a class="ref" href="functor0.html#D35" title="FUNCTOR0:def.35">FUNCTOR0:def 35</a>,<a class="ref" href="functor0.html#D36" title="FUNCTOR0:def.36">FUNCTOR0:def 36</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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