<?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  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="altcat_1.html#V2" title="ALTCAT_1:attr.2">transitive</a> <a href="altcat_1.html#V12" title="ALTCAT_1:attr.12">with_units</a>  <a href="altcat_1.html#L2" title="ALTCAT_1:struct.2">AltCatStr</a> ;<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>;<br/>



<b>assume </b><a NAME="E1:122"/><i><font color="Green" title="E81">A1</font></i>: 
<font color="Maroon" title="c3">F</font> is <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a>
 ;<br/>

<a NAME="E2:122"/><b>then </b>
<font color="Maroon" title="c3">F</font> is <a href="functor0.html#V19" title="FUNCTOR0:attr.19">injective</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D36" target="_self" title="FUNCTOR0:def.36">Def36</a></i>;<br/>
<a NAME="E3:122"/><b>then </b>
<font color="Maroon" title="c3">F</font> is <a href="functor0.html#V4" title="FUNCTOR0:attr.4">one-to-one</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D34" target="_self" title="FUNCTOR0:def.34">Def34</a></i>;<br/>
<a NAME="E4:122"/><b>then </b><i><font color="Green" title="E82">A2</font></i>: 
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>
 
<b>by </b><i><a class="ref" href="functor0.html#D7" target="_self" title="FUNCTOR0:def.7">Def7</a></i>;<br/>
<a NAME="E5:122"/><i><font color="Green" title="E83">A3</font></i>: 
<font color="Maroon" title="c3">F</font> is <a href="functor0.html#V8" title="FUNCTOR0:attr.8">feasible</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D26" target="_self" title="FUNCTOR0:def.26">Def26</a></i>;<br/>
<a NAME="E6:122"/><b>then </b><i><font color="Green" title="E84">A4</font></i>: 
( <font color="Maroon" title="c3">F</font> <a href="functor0.html#K15" title="FUNCTOR0:func.15">"</a>  is <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a> &amp; <font color="Maroon" title="c3">F</font> <a href="functor0.html#K15" title="FUNCTOR0:func.15">"</a>  is <a href="functor0.html#V8" title="FUNCTOR0:attr.8">feasible</a> )
 
<b>by </b><i><a class="txt" href="functor0.html#E1:122"><i><font color="Green" title="E81">A1</font></i></a>, <a class="ref" href="functor0.html#T36" target="_self" title="FUNCTOR0:th.36">Th36</a></i>;<br/>
<a NAME="E7:122"/><i><font color="Green" title="E85">A5</font></i>: 
<font color="Maroon" title="c3">F</font> is <a href="functor0.html#V12" title="FUNCTOR0:attr.12">id-preserving</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D26" target="_self" title="FUNCTOR0:def.26">Def26</a></i>;<br/>
<a NAME="E8:122"/><i><font color="Green" title="E86">A6</font></i>: 
<font color="Maroon" title="c3">F</font> is <a href="functor0.html#V14" title="FUNCTOR0:attr.14">comp-reversing</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D28" target="_self" title="FUNCTOR0:def.28">Def28</a></i>;<br/>
<a NAME="E9:122"/>
<font color="Maroon" title="c3">F</font> is <a href="functor0.html#V20" title="FUNCTOR0:attr.20">surjective</a>
 
<b>by </b><i><a class="txt" href="functor0.html#E1:122"><i><font color="Green" title="E81">A1</font></i></a>, <a class="ref" href="functor0.html#D36" target="_self" title="FUNCTOR0:def.36">Def36</a></i>;<br/>
<a NAME="E10:122"/><b>then </b><i><font color="Green" title="E87">A7</font></i>: 
<font color="Maroon" title="c3">F</font> is <a href="functor0.html#V5" title="FUNCTOR0:attr.5">onto</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D35" target="_self" title="FUNCTOR0:def.35">Def35</a></i>;<br/>
<a NAME="E11:122"/><b>then </b><i><font color="Green" title="E88">A8</font></i>: 
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="ref" href="functor0.html#D8" target="_self" title="FUNCTOR0:def.8">Def8</a></i>;<br/>
<a NAME="E12:122"/><i><font color="Green" title="E89">A9</font></i>: 
<font color="Maroon" title="c3">F</font> is <a href="functor0.html#V11" title="FUNCTOR0:attr.11">Contravariant</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D28" target="_self" title="FUNCTOR0:def.28">Def28</a></i>;<br/>
<a NAME="E13:122"/><i><font color="Green" title="E90">A10</font></i>: 
<font color="Maroon" title="c3">F</font> is <a href="functor0.html#V7" title="FUNCTOR0:attr.7">coreflexive</a>
 
<b>by </b><i><a class="txt" href="functor0.html#E10:122"><i><font color="Green" title="E87">A7</font></i></a>, <a class="ref" href="functor0.html#T48" target="_self" title="FUNCTOR0:th.48">Th48</a></i>;<br/>
<a NAME="E14:122"/><i><font color="Green" title="E91">A11</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="functor0.html#K15" title="FUNCTOR0:func.15">"</a>  is <a href="functor0.html#V11" title="FUNCTOR0:attr.11">Contravariant</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1"><b>proof </b></a><div class="add">

<a NAME="E1:122_1"/>
<font color="Maroon" title="c3">F</font> is <a href="functor0.html#V11" title="FUNCTOR0:attr.11">Contravariant</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D28" target="_self" title="FUNCTOR0:def.28">Def28</a></i>;<br/>
<a NAME="E2:122_1"/><b>then </b>
the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c3">F</font> is <a href="functor0.html#V2" title="FUNCTOR0:attr.2">Contravariant</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D14" target="_self" title="FUNCTOR0:def.14">Def14</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font><b> such that </b><br/><a NAME="E4:122_1"/><i><font color="Green" title="E92">A12</font></i>: 
the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c3">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="functor0.html#K1" title="FUNCTOR0:func.1">~</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c4">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span>
 <b>by </b><i><a class="ref" href="functor0.html#D3" target="_self" title="FUNCTOR0:def.3">Def3</a></i>;<br/>
<a NAME="E5:122_1"/>
<span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c4">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="functor0.html#E4:122"><i><font color="Green" title="E82">A2</font></i></a>, <a class="txt" href="functor0.html#E4:122_1"><i><font color="Green" title="E92">A12</font></i></a>, <a class="ref" href="functor0.html#T10" target="_self" title="FUNCTOR0:th.10">Th10</a></i>;<br/>
<a NAME="E6:122_1"/><b>then </b><i><font color="Green" title="E93">A13</font></i>: 
<font color="Maroon" title="c4">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="functor0.html#T8" target="_self" title="FUNCTOR0:th.8">Th8</a></i>;<br/>
<a NAME="E7:122_1"/><b>then </b><i><font color="Green" title="E94">A14</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/>
<a NAME="E8:122_1"/>
<span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c4">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 
<b>by </b><i><a class="txt" href="functor0.html#E11:122"><i><font color="Green" title="E88">A8</font></i></a>, <a class="txt" href="functor0.html#E4:122_1"><i><font color="Green" title="E92">A12</font></i></a>, <a class="ref" href="functor0.html#T14" target="_self" title="FUNCTOR0:th.14">Th14</a></i>;<br/>
<a NAME="E9:122_1"/><b>then </b><i><font color="Green" title="E95">A15</font></i>: 
 <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">rng</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c4">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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 <font color="Maroon" title="c2">B</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E10:122_1"/>
 <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">rng</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c4">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_3.html#T88" title="FUNCT_3:th.88">FUNCT_3:88</a></i>;<br/>
<a NAME="E11:122_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> <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="c2">B</font>
 
<b>by </b><i><a class="txt" href="functor0.html#E9:122_1"><i><font color="Green" title="E95">A15</font></i></a>, <a class="ref" href="zfmisc_1.html#T115" title="ZFMISC_1:th.115">ZFMISC_1:115</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="functor0.html#E7:122_1"><i><font color="Green" title="E94">A14</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">g</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D3" target="_self" title="FUNCTOR0:def.3">FUNCTOR0:def 3</a>,<a class="ref" href="functor0.html#D14" target="_self" title="FUNCTOR0:def.14">FUNCTOR0:def 14</a><br/>

<a NAME="E13:122_1"/><i><font color="Green" title="E96">A16</font></i>: 
<span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c4">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c5">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span>
 
<b>by </b><i><a class="txt" href="functor0.html#E6:122_1"><i><font color="Green" title="E93">A13</font></i></a>, <a class="ref" href="functor0.html#T7" target="_self" title="FUNCTOR0:th.7">Th7</a></i>;<br/>
<b>thus </b>the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="functor0.html#K15" title="FUNCTOR0:func.15">"</a> </span>)</span> = 
the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
<b>by </b><i><a class="txt" href="functor0.html#E1:122"><i><font color="Green" title="E81">A1</font></i></a>, <a class="ref" href="functor0.html#D39" target="_self" title="FUNCTOR0:def.39">Def39</a></i>
<br/>.= 
 <a href="functor0.html#K1" title="FUNCTOR0:func.1">~</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c5">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span>
<b>by </b><i><a class="txt" href="functor0.html#E4:122_1"><i><font color="Green" title="E92">A12</font></i></a>, <a class="txt" href="functor0.html#E6:122_1"><i><font color="Green" title="E93">A13</font></i></a>, <a class="txt" href="functor0.html#E13:122_1"><i><font color="Green" title="E96">A16</font></i></a>, <a class="ref" href="functor0.html#T11" target="_self" title="FUNCTOR0:th.11">Th11</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E15:122"/><i><font color="Green" title="E92">A17</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="functor0.html#K15" title="FUNCTOR0:func.15">"</a>  is <a href="functor0.html#V12" title="FUNCTOR0:attr.12">id-preserving</a>
 
<b>by </b><i><a class="txt" href="functor0.html#E1:122"><i><font color="Green" title="E81">A1</font></i></a>, <a class="txt" href="functor0.html#E5:122"><i><font color="Green" title="E83">A3</font></i></a>, <a class="txt" href="functor0.html#E7:122"><i><font color="Green" title="E85">A5</font></i></a>, <a class="txt" href="functor0.html#E13:122"><i><font color="Green" title="E90">A10</font></i></a>, <a class="ref" href="functor0.html#T38" target="_self" title="FUNCTOR0:th.38">Th38</a></i>;<br/>
<a NAME="E16:122"/>
<font color="Maroon" title="c3">F</font> <a href="functor0.html#K15" title="FUNCTOR0:func.15">"</a>  is <a href="functor0.html#V14" title="FUNCTOR0:attr.14">comp-reversing</a>
 
<b>by </b><i><a class="txt" href="functor0.html#E1:122"><i><font color="Green" title="E81">A1</font></i></a>, <a class="txt" href="functor0.html#E5:122"><i><font color="Green" title="E83">A3</font></i></a>, <a class="txt" href="functor0.html#E8:122"><i><font color="Green" title="E86">A6</font></i></a>, <a class="txt" href="functor0.html#E12:122"><i><font color="Green" title="E89">A9</font></i></a>, <a class="txt" href="functor0.html#E13:122"><i><font color="Green" title="E90">A10</font></i></a>, <a class="ref" href="functor0.html#T44" target="_self" title="FUNCTOR0:th.44">Th44</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">G</font> = <font color="Maroon" title="c3">F</font> <a href="functor0.html#K15" title="FUNCTOR0:func.15">"</a>  as    <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="functor0.html#E6:122"><i><font color="Green" title="E84">A4</font></i></a>, <a class="txt" href="functor0.html#E14:122"><i><font color="Green" title="E91">A11</font></i></a>, <a class="txt" href="functor0.html#E15:122"><i><font color="Green" title="E92">A17</font></i></a>, <a class="ref" href="functor0.html#D26" target="_self" title="FUNCTOR0:def.26">Def26</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">G</font>
;<br/>

<b>thus </b><a NAME="E18:122"/>
<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="functor0.html#K15" title="FUNCTOR0:func.15">"</a> 
 ;<br/>

<b>thus </b><a NAME="E19:122"/>
<font color="Maroon" title="c4">G</font> is <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a>
 <b>by </b><i><a class="txt" href="functor0.html#E1:122"><i><font color="Green" title="E81">A1</font></i></a>, <a class="txt" href="functor0.html#E5:122"><i><font color="Green" title="E83">A3</font></i></a>, <a class="ref" href="functor0.html#T36" target="_self" title="FUNCTOR0:th.36">Th36</a></i>;<br/>

<b>thus </b><a NAME="E20:122"/>
<font color="Maroon" title="c4">G</font> is <a href="functor0.html#V11" title="FUNCTOR0:attr.11">Contravariant</a>
 <b>by </b><i><a class="txt" href="functor0.html#E14:122"><i><font color="Green" title="E91">A11</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D28" target="_self" title="FUNCTOR0:def.28">FUNCTOR0:def 28</a><br/>

<b>thus </b><a NAME="E21:122"/>
<font color="Maroon" title="c4">G</font> is <a href="functor0.html#V14" title="FUNCTOR0:attr.14">comp-reversing</a>
 <b>by </b><i><a class="txt" href="functor0.html#E1:122"><i><font color="Green" title="E81">A1</font></i></a>, <a class="txt" href="functor0.html#E5:122"><i><font color="Green" title="E83">A3</font></i></a>, <a class="txt" href="functor0.html#E8:122"><i><font color="Green" title="E86">A6</font></i></a>, <a class="txt" href="functor0.html#E12:122"><i><font color="Green" title="E89">A9</font></i></a>, <a class="txt" href="functor0.html#E13:122"><i><font color="Green" title="E90">A10</font></i></a>, <a class="ref" href="functor0.html#T44" target="_self" title="FUNCTOR0:th.44">Th44</a></i>;<br/>


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