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

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="Olive" title="b1">F</font> is  <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a>  holds <br/> ex <font color="Olive" title="b2">G</font> being   <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> st <br/>( <font color="Olive" title="b2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="functor0.html#K15" title="FUNCTOR0:func.15">"</a>  &amp; <font color="Olive" title="b2">G</font> is  <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a>  &amp; <font color="Olive" title="b2">G</font> is  <a href="functor0.html#V16" title="FUNCTOR0:attr.16">contravariant</a>  )</span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> is  <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a>  implies  ex <font color="Olive" title="b1">G</font> being   <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> st <br/>( <font color="Olive" title="b1">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>  &amp; <font color="Olive" title="b1">G</font> is  <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a>  &amp; <font color="Olive" title="b1">G</font> is  <a href="functor0.html#V16" title="FUNCTOR0:attr.16">contravariant</a>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:123"/><span class="lab"><font color="Green" title="E81">A1</font></span>: 
<font color="Maroon" title="c3">F</font> is  <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">G</font> being   <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> st <br/>( <font color="Olive" title="b1">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>  &amp; <font color="Olive" title="b1">G</font> is  <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a>  &amp; <font color="Olive" title="b1">G</font> is  <a href="functor0.html#V16" title="FUNCTOR0:attr.16">contravariant</a>  )</span><br/>

<a NAME="E2:123"/><span class="kw">then </span>
<font color="Maroon" title="c3">F</font> is  <a href="functor0.html#V19" title="FUNCTOR0:attr.19">injective</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D36" target="_self" title="FUNCTOR0:def.36">Def36</a></span>;<br/>
<a NAME="E3:123"/><span class="kw">then </span>
<font color="Maroon" title="c3">F</font> is  <a href="functor0.html#V4" title="FUNCTOR0:attr.4">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D34" target="_self" title="FUNCTOR0:def.34">Def34</a></span>;<br/>
<a NAME="E4:123"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A2</font></span>: 
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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D7" target="_self" title="FUNCTOR0:def.7">Def7</a></span>;<br/>
<a NAME="E5:123"/><span class="lab"><font color="Green" title="E83">A3</font></span>: 
<font color="Maroon" title="c3">F</font> is  <a href="functor0.html#V8" title="FUNCTOR0:attr.8">feasible</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D26" target="_self" title="FUNCTOR0:def.26">Def26</a></span>;<br/>
<a NAME="E6:123"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A4</font></span>: 
( <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>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:123"><span class="lab"><font color="Green" title="E81">A1</font></span></a>, <a class="ref" href="functor0.html#T36" target="_self" title="FUNCTOR0:th.36">Th36</a></span>;<br/>
<a NAME="E7:123"/><span class="lab"><font color="Green" title="E85">A5</font></span>: 
<font color="Maroon" title="c3">F</font> is  <a href="functor0.html#V12" title="FUNCTOR0:attr.12">id-preserving</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D26" target="_self" title="FUNCTOR0:def.26">Def26</a></span>;<br/>
<a NAME="E8:123"/><span class="lab"><font color="Green" title="E86">A6</font></span>: 
<font color="Maroon" title="c3">F</font> is  <a href="functor0.html#V14" title="FUNCTOR0:attr.14">comp-reversing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D28" target="_self" title="FUNCTOR0:def.28">Def28</a></span>;<br/>
<a NAME="E9:123"/>
<font color="Maroon" title="c3">F</font> is  <a href="functor0.html#V20" title="FUNCTOR0:attr.20">surjective</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:123"><span class="lab"><font color="Green" title="E81">A1</font></span></a>, <a class="ref" href="functor0.html#D36" target="_self" title="FUNCTOR0:def.36">Def36</a></span>;<br/>
<a NAME="E10:123"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A7</font></span>: 
<font color="Maroon" title="c3">F</font> is  <a href="functor0.html#V5" title="FUNCTOR0:attr.5">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D35" target="_self" title="FUNCTOR0:def.35">Def35</a></span>;<br/>
<a NAME="E11:123"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A8</font></span>: 
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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D8" target="_self" title="FUNCTOR0:def.8">Def8</a></span>;<br/>
<a NAME="E12:123"/><span class="lab"><font color="Green" title="E89">A9</font></span>: 
<font color="Maroon" title="c3">F</font> is  <a href="functor0.html#V11" title="FUNCTOR0:attr.11">Contravariant</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D28" target="_self" title="FUNCTOR0:def.28">Def28</a></span>;<br/>
<a NAME="E13:123"/><span class="lab"><font color="Green" title="E90">A10</font></span>: 
<font color="Maroon" title="c3">F</font> is  <a href="functor0.html#V7" title="FUNCTOR0:attr.7">coreflexive</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:123"><span class="lab"><font color="Green" title="E87">A7</font></span></a>, <a class="ref" href="functor0.html#T48" target="_self" title="FUNCTOR0:th.48">Th48</a></span>;<br/>
<a NAME="E14:123"/><span class="lab"><font color="Green" title="E91">A11</font></span>: 
<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="123_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:123_1"/>
<font color="Maroon" title="c3">F</font> is  <a href="functor0.html#V11" title="FUNCTOR0:attr.11">Contravariant</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D28" target="_self" title="FUNCTOR0:def.28">Def28</a></span>;<br/>
<a NAME="E2:123_1"/><span class="kw">then </span>
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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D14" target="_self" title="FUNCTOR0:def.14">Def14</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:123_1"/><span class="lab"><font color="Green" title="E92">A12</font></span>: 
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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D3" target="_self" title="FUNCTOR0:def.3">Def3</a></span>;<br/>
<a NAME="E5:123_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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:123"><span class="lab"><font color="Green" title="E82">A2</font></span></a>, <a class="txt" href="#E4:123_1"><span class="lab"><font color="Green" title="E92">A12</font></span></a>, <a class="ref" href="functor0.html#T10" target="_self" title="FUNCTOR0:th.10">Th10</a></span>;<br/>
<a NAME="E6:123_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A13</font></span>: 
<font color="Maroon" title="c4">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#T8" target="_self" title="FUNCTOR0:th.8">Th8</a></span>;<br/>
<a NAME="E7:123_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A14</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></span>;<br/>
<a NAME="E8:123_1"/><span class="lab"><font color="Green" title="E95">A15</font></span>: 
 <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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:123_1"><span class="lab"><font color="Green" title="E93">A13</font></span></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></span>;<br/>
<a NAME="E9:123_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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:123"><span class="lab"><font color="Green" title="E88">A8</font></span></a>, <a class="txt" href="#E4:123_1"><span class="lab"><font color="Green" title="E92">A12</font></span></a>, <a class="ref" href="functor0.html#T14" target="_self" title="FUNCTOR0:th.14">Th14</a></span>;<br/>
<a NAME="E10:123_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A16</font></span>: 
 <a href="funcop_1.html#K11" title="FUNCOP_1:func.11">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E11:123_1"/>
 <a href="funcop_1.html#K11" title="FUNCOP_1:func.11">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="relset_1.html#R2" title="RELSET_1:pred.2">=</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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#T88" title="FUNCT_3:th.88">FUNCT_3:88</a></span>;<br/>
<a NAME="E12:123_1"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:123_1"><span class="lab"><font color="Green" title="E96">A16</font></span></a>, <a class="ref" href="zfmisc_1.html#T115" title="ZFMISC_1:th.115">ZFMISC_1:115</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:123_1"><span class="lab"><font color="Green" title="E94">A14</font></span></a>, <a class="txt" href="#E8:123_1"><span class="lab"><font color="Green" title="E95">A15</font></span></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></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">g</font>
; <span class="comment"><font color="Red">:: according to </font></span><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> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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> <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="c5">g</font>,<font color="Maroon" title="c5">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span><br/>

<a NAME="E14:123_1"/><span class="lab"><font color="Green" title="E97">A17</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:123_1"><span class="lab"><font color="Green" title="E93">A13</font></span></a>, <a class="ref" href="functor0.html#T7" target="_self" title="FUNCTOR0:th.7">Th7</a></span>;<br/>
<span class="kw">thus </span>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> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:123"><span class="lab"><font color="Green" title="E81">A1</font></span></a>, <a class="ref" href="functor0.html#D39" target="_self" title="FUNCTOR0:def.39">Def39</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:123_1"><span class="lab"><font color="Green" title="E92">A12</font></span></a>, <a class="txt" href="#E6:123_1"><span class="lab"><font color="Green" title="E93">A13</font></span></a>, <a class="txt" href="#E14:123_1"><span class="lab"><font color="Green" title="E97">A17</font></span></a>, <a class="ref" href="functor0.html#T11" target="_self" title="FUNCTOR0:th.11">Th11</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E15:123"/><span class="lab"><font color="Green" title="E92">A18</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:123"><span class="lab"><font color="Green" title="E81">A1</font></span></a>, <a class="txt" href="#E5:123"><span class="lab"><font color="Green" title="E83">A3</font></span></a>, <a class="txt" href="#E7:123"><span class="lab"><font color="Green" title="E85">A5</font></span></a>, <a class="txt" href="#E13:123"><span class="lab"><font color="Green" title="E90">A10</font></span></a>, <a class="ref" href="functor0.html#T38" target="_self" title="FUNCTOR0:th.38">Th38</a></span>;<br/>
<a NAME="E16:123"/>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:123"><span class="lab"><font color="Green" title="E81">A1</font></span></a>, <a class="txt" href="#E5:123"><span class="lab"><font color="Green" title="E83">A3</font></span></a>, <a class="txt" href="#E8:123"><span class="lab"><font color="Green" title="E86">A6</font></span></a>, <a class="txt" href="#E12:123"><span class="lab"><font color="Green" title="E89">A9</font></span></a>, <a class="txt" href="#E13:123"><span class="lab"><font color="Green" title="E90">A10</font></span></a>, <a class="ref" href="functor0.html#T44" target="_self" title="FUNCTOR0:th.44">Th44</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:123"><span class="lab"><font color="Green" title="E84">A4</font></span></a>, <a class="txt" href="#E14:123"><span class="lab"><font color="Green" title="E91">A11</font></span></a>, <a class="txt" href="#E15:123"><span class="lab"><font color="Green" title="E92">A18</font></span></a>, <a class="ref" href="functor0.html#D26" target="_self" title="FUNCTOR0:def.26">Def26</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">G</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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>  &amp; <font color="Maroon" title="c4">G</font> is  <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a>  &amp; <font color="Maroon" title="c4">G</font> is  <a href="functor0.html#V16" title="FUNCTOR0:attr.16">contravariant</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E18:123"/>
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">G</font> is  <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a>  &amp; <font color="Maroon" title="c4">G</font> is  <a href="functor0.html#V16" title="FUNCTOR0:attr.16">contravariant</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E19:123"/>
<font color="Maroon" title="c4">G</font> is  <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:123"><span class="lab"><font color="Green" title="E81">A1</font></span></a>, <a class="txt" href="#E5:123"><span class="lab"><font color="Green" title="E83">A3</font></span></a>, <a class="ref" href="functor0.html#T36" target="_self" title="FUNCTOR0:th.36">Th36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">G</font> is  <a href="functor0.html#V16" title="FUNCTOR0:attr.16">contravariant</a> </span><br/>

<span class="kw">thus </span><a NAME="E20:123"/>
<font color="Maroon" title="c4">G</font> is  <a href="functor0.html#V11" title="FUNCTOR0:attr.11">Contravariant</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:123"><span class="lab"><font color="Green" title="E91">A11</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="functor0.html#D28" target="_self" title="FUNCTOR0:def.28">FUNCTOR0:def 28</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">G</font> is  <a href="functor0.html#V14" title="FUNCTOR0:attr.14">comp-reversing</a> </span><br/>

<span class="kw">thus </span><a NAME="E21:123"/>
<font color="Maroon" title="c4">G</font> is  <a href="functor0.html#V14" title="FUNCTOR0:attr.14">comp-reversing</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:123"><span class="lab"><font color="Green" title="E81">A1</font></span></a>, <a class="txt" href="#E5:123"><span class="lab"><font color="Green" title="E83">A3</font></span></a>, <a class="txt" href="#E8:123"><span class="lab"><font color="Green" title="E86">A6</font></span></a>, <a class="txt" href="#E12:123"><span class="lab"><font color="Green" title="E89">A9</font></span></a>, <a class="txt" href="#E13:123"><span class="lab"><font color="Green" title="E90">A10</font></span></a>, <a class="ref" href="functor0.html#T44" target="_self" title="FUNCTOR0:th.44">Th44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
