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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_1_1"/>
( <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_1"><b>suppose </b></a><a NAME="E1:22_1_1_1_1"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:22_1_1_1_1"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T113" title="ZFMISC_1:th.113">ZFMISC_1:113</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c3">f</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="functor0.html#NM1" title="FUNCTOR0:NM.1">bifunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <b>by </b><i><a class="ref" href="functor0.html#T1" target="_self" title="FUNCTOR0:th.1">Th1</a></i>;<br/><b>take </b>
<font color="Maroon" title="c3">f</font>
;<br/><b>reconsider </b><font color="Maroon" title="c4">g</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <b>by </b><i><a class="txt" href="functor0.html#E1:22_1_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="functor0.html#T1" target="_self" title="FUNCTOR0:th.1">Th1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">h</font> = <font color="Maroon" title="c4">g</font> as  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/><b>thus </b><a NAME="E6:22_1_1_1_1"/>
<font color="Maroon" title="c3">f</font> is <a href="functor0.html#V1" title="FUNCTOR0:attr.1">Covariant</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c4">g</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D2" target="_self" title="FUNCTOR0:def.2">FUNCTOR0:def 2</a><br/>

<b>thus </b><font color="Maroon" title="c3">f</font> = 
<span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c5">h</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span>

<br/>.= 
<span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c4">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span>

;<br/>


</div><b>end;</b></div><b>take </b>
<font color="Maroon" title="c4">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><br/><b>thus </b><font color="Maroon" title="c3">f</font> = 
 <a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c5">h</font>,<font color="Maroon" title="c5">h</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</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="c4">g</font>,<font color="Maroon" title="c4">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span>

;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_2"><b>suppose </b></a><a NAME="E1:22_1_1_1_2"/><i><font color="Green" title="E18">A2</font></i>: 
<font color="Maroon" title="c2">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c3">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">B</font>;<br/><b>set </b><font color="Maroon" title="c4">f</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="pboole.html#K9" title="PBOOLE:func.9">--&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/><a NAME="E3:22_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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"><font color="Maroon" title="c2">B</font>,<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="txt" href="functor0.html#E1:22_1_1_1_2"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c5">f</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="pboole.html#K9" title="PBOOLE:func.9">--&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="functor0.html#NM1" title="FUNCTOR0:NM.1">bifunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <b>by </b><i><a class="ref" href="funcop_1.html#T57" title="FUNCOP_1:th.57">FUNCOP_1:57</a></i>;<br/><b>take </b>
<font color="Maroon" title="c5">f</font>
;<br/><b>thus </b><a NAME="E5:22_1_1_1_2"/>
( <font color="Maroon" title="c5">f</font> is <a href="functor0.html#V1" title="FUNCTOR0:attr.1">Covariant</a> &amp; <font color="Maroon" title="c5">f</font> is <a href="functor0.html#V2" title="FUNCTOR0:attr.2">Contravariant</a> )
 <b>by </b><i><a class="txt" href="functor0.html#E1:22_1_1_1_2"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="functor0.html#T16" target="_self" title="FUNCTOR0:th.16">Th16</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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