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<b>let </b><font color="Maroon">A</font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V2">transitive</a> <a href="altcat_1.html#V12">with_units</a>  <a href="altcat_1.html#L2">AltCatStr</a> , <font color="Maroon">B</font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V2">transitive</a> <a href="altcat_1.html#V12">with_units</a>  <a href="altcat_1.html#L2">AltCatStr</a> ;<br/><b>let </b><font color="Maroon">F</font> be  <a href="functor0.html#V8" target="_self">feasible</a>  <a href="functor0.html#L2" target="_self">FunctorStr</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/>



<b>assume </b><a NAME="E1:108"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">F</font> is <a href="functor0.html#V21" target="_self">bijective</a> &amp; <font color="Maroon">F</font> is <a href="functor0.html#V11" target="_self">Contravariant</a> )
 ;<br/>

<a NAME="E2:108"/><b>then </b>
<font color="Maroon">F</font> is <a href="functor0.html#V19" target="_self">injective</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:108"/><b>then </b>
<font color="Maroon">F</font> is <a href="functor0.html#V4" target="_self">one-to-one</a>
 
<b>by </b><i><a class="ref" href="functor0.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E4:108"/><b>then </b><i><font color="Green">E23</font></i>: 
the <a href="functor0.html#U1" target="_self">ObjectMap</a> of <font color="Maroon">F</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="functor0.html#E4:108"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E5:108"/>
<font color="Maroon">F</font> is <a href="functor0.html#V20" target="_self">surjective</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:108"/><b>then </b>
<font color="Maroon">F</font> is <a href="functor0.html#V5" target="_self">onto</a>
 
<b>by </b><i/>;<br/>
<a NAME="E7:108"/><b>then </b><i><font color="Green">E26</font></i>: 
the <a href="functor0.html#U1" target="_self">ObjectMap</a> of <font color="Maroon">F</font> is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i/>;<br/>
<a NAME="E8:108"/>
the <a href="functor0.html#U1" target="_self">ObjectMap</a> of <font color="Maroon">F</font> is <a href="functor0.html#V2" target="_self">Contravariant</a>
 
<b>by </b><i>, <a class="txt" href="functor0.html#E12:108"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E10:108"/><i><font color="Green">E30</font></i>: 
the <a href="functor0.html#U1" target="_self">ObjectMap</a> of <font color="Maroon">F</font> <a href="funct_2.html#R2">=</a>  <a href="functor0.html#K1" target="_self">~</a> <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">f</font></span><a href="funct_3.html#K16">:]</a></span>
 <b>by </b><i/>;<br/>
<a NAME="E11:108"/>
<span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">f</font></span><a href="funct_3.html#K16">:]</a></span> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E12:108"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i/>;<br/>
<a NAME="E13:108"/><b>then </b><i><font color="Green">E33</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<a NAME="E14:108"/>
<span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">f</font></span><a href="funct_3.html#K16">:]</a></span> is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E15:108"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="funcop_1.html#K9">rng</a> <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">f</font></span><a href="funct_3.html#K16">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E16:108"/>
 <a href="funcop_1.html#K9">rng</a> <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">f</font></span><a href="funct_3.html#K16">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_3.html#T88">FUNCT_3:88</a></i>;<br/>
<a NAME="E17:108"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T115">ZFMISC_1:115</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="funct_1.html#K2">"</a>  as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>take </b>
<font color="Maroon">g</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D3" target="_self">FUNCTOR0:def 3</a>,<a class="ref" href="functor0.html#D14" target="_self">FUNCTOR0:def 14</a><br/>

<a NAME="E19:108"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">f</font></span><a href="funct_3.html#K16">:]</a></span> <a href="funct_1.html#K2">"</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K16">:]</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>thus </b>the <a href="functor0.html#U1" target="_self">ObjectMap</a> of <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K15" target="_self">"</a> </span>)</span> = 
the <a href="functor0.html#U1" target="_self">ObjectMap</a> of <font color="Maroon">F</font> <a href="funct_1.html#K2">"</a> 
<b>by </b><i>, </i>
<br/>.= 
 <a href="functor0.html#K1" target="_self">~</a> <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K16">:]</a></span>
<b>by </b><i>, , , </i>
;<br/>


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