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<b>reconsider </b><font color="Maroon">OM</font> =  <a href="partfun1.html#K6">id</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">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> as   <a href="functor0.html#NM1">bifunction</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">A</font> ;<br/>
<b>set </b><font color="Maroon">MM</font> =  <a href="msualg_3.html#K2">id</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font>;<br/>
<a NAME="E2:73_1_1"/>
 <a href="msualg_3.html#K2">id</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> is    <a href="functor0.html#M1" target="_self">MSUnTrans</a> of <font color="Maroon">OM</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:73_1_1_1_1"/>
( <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">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <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">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_1_1_1"><b>case </b></a><a NAME="E1:73_1_1_1_1_1"/>
<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">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">I2'</font> = <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">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>reconsider </b><font color="Maroon">A'</font> = the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I2'</font> ;<br/><b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">OM</font> as   <a href="funct_2.html#NM1">Function</a> of <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">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">I2'</font> ;<br/><b>take </b>
<font color="Maroon">I2'</font>
;<br/><b>take </b>
<font color="Maroon">A'</font>
;<br/><b>take </b>
<font color="Maroon">f'</font>
;<br/><b>thus </b><a NAME="E5:73_1_1_1_1_1"/>
( <font color="Maroon">OM</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> &amp; the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font> )
 ;<br/><b>thus </b><a NAME="E6:73_1_1_1_1_1"/>
 <a href="msualg_3.html#K2">id</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> is    <a href="pboole.html#M3">ManySortedFunction</a> of the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font>,<font color="Maroon">A'</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a><br/>

<b>assume </b><a NAME="E1:73_1_1_1_1_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</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">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>then </b><span class="p1">(<span class="default"><font color="Maroon">A'</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">A'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="functor0.html#T6" target="_self">Th6</a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/>
<b>hence </b><a NAME="E3:73_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="funct_2.html#NM1">Function</a> of the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">A'</font> <a href="pboole.html#K7">*</a> <font color="Maroon">f'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="functor0.html#T6" target="_self">Th6</a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_1_1_2"><b>case </b></a><a NAME="E1:73_1_1_1_1_2"/><i><font color="Green">E23</font></i>: 
<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">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:73_1_1_1_1_2"/><b>then </b>
 <a href="msualg_3.html#K2">id</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="pboole.html#T134">PBOOLE:134</a></i>;<br/><b>hence </b><a NAME="E3:73_1_1_1_1_2"/>
 <a href="msualg_3.html#K2">id</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> <a href="pboole.html#R6">=</a>  <a href="pboole.html#K1">[0]</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">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="functor0.html#T6" target="_self">Th6</a>, <a class="ref" href="pboole.html#T134">PBOOLE:134</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">MM</font> =  <a href="msualg_3.html#K2">id</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> as    <a href="functor0.html#M1" target="_self">MSUnTrans</a> of <font color="Maroon">OM</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> ;<br/>
<b>take </b>
 <a href="functor0.html#G2" target="_self">FunctorStr</a>(# <font color="Maroon">OM</font>,<font color="Maroon">MM</font> #)
;<br/>

<b>thus </b><a NAME="E4:73_1_1"/>
( the <a href="functor0.html#U1" target="_self">ObjectMap</a> of <a href="functor0.html#G2" target="_self">FunctorStr</a>(# <font color="Maroon">OM</font>,<font color="Maroon">MM</font> #) <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</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">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; the <a href="functor0.html#U2" target="_self">MorphMap</a> of <a href="functor0.html#G2" target="_self">FunctorStr</a>(# <font color="Maroon">OM</font>,<font color="Maroon">MM</font> #) <a href="pboole.html#R6">=</a>  <a href="msualg_3.html#K2">id</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> )
 ;<br/>


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