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<div class="add">

<b>let </b><font color="Maroon" title="c1">I1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">I2</font>, <font color="Maroon" title="c3">I3</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">I1</font>,<font color="Maroon" title="c2">I2</font>;<br/><b>let </b><font color="Maroon" title="c5">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">I2</font>,<font color="Maroon" title="c3">I3</font>;<br/><b>let </b><font color="Maroon" title="c6">A</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I1</font>;<br/><b>let </b><font color="Maroon" title="c7">B</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c2">I2</font>;<br/><b>let </b><font color="Maroon" title="c8">C</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c3">I3</font>;<br/><b>let </b><font color="Maroon" title="c9">F</font> be    <a href="functor0.html#M1" title="FUNCTOR0:mode.1">MSUnTrans</a> of <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">B</font>;<br/><b>let </b><font color="Maroon" title="c10">G</font> be    <a href="functor0.html#M1" title="FUNCTOR0:mode.1">MSUnTrans</a> of <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c7">B</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c8">C</font>;<br/>

















<b>assume </b><a NAME="E1:30"/><i><font color="Green" title="E23">A1</font></i>: 
for <font color="Olive" title="b1">ii</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">ii</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">ii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; not <font color="Maroon" title="c6">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">ii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c8">C</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">ii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c11">G</font> = <font color="Maroon" title="c10">G</font> as    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c7">B</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c8">C</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <b>by </b><i><a class="ref" href="functor0.html#D5" target="_self" title="FUNCTOR0:def.5">Def5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">F</font> = <font color="Maroon" title="c9">F</font> as    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">B</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font> <b>by </b><i><a class="ref" href="functor0.html#D5" target="_self" title="FUNCTOR0:def.5">Def5</a></i>;<br/>
<a NAME="E4:30"/><i><font color="Green" title="E24">A2</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c11">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I1</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I1</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<i><font color="Green" title="E25">A3</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="c11">G</font> <a href="altcat_2.html#K3" title="ALTCAT_2:func.3">**</a> <font color="Maroon" title="c12">F</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c11">G</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">F</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D24" title="PBOOLE:def.24">PBOOLE:def 24</a></i>
<br/>.= 
<font color="Maroon" title="c1">I1</font>
<b>by </b><i><a class="txt" href="functor0.html#E4:30"><i><font color="Green" title="E24">A2</font></i></a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c13">GF</font> = <font color="Maroon" title="c11">G</font> <a href="altcat_2.html#K3" title="ALTCAT_2:func.3">**</a> <font color="Maroon" title="c12">F</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I1</font> ;<br/>
<a NAME="E7:30"/>
<font color="Maroon" title="c13">GF</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c8">C</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">ii</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a><br/>

<b>assume </b><a NAME="E1:30_2"/><i><font color="Green" title="E26">A4</font></i>: 
<font color="Maroon" title="c14">ii</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c15">Fi</font> = <font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">ii</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c6">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">ii</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">ii</font> <b>by </b><i><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">Gi</font> = <font color="Maroon" title="c11">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">ii</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">ii</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">C</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">ii</font> <b>by </b><i><a class="txt" href="functor0.html#E1:30_2"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/>
<a NAME="E4:30_2"/>
(  not <span class="p1">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">ii</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="c6">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">ii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c8">C</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">ii</font> <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="txt" href="functor0.html#E1:30"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="functor0.html#E1:30_2"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/>
<a NAME="E5:30_2"/><b>then </b>
<font color="Maroon" title="c16">Gi</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c15">Fi</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c6">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">ii</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">C</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">ii</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T19" title="FUNCT_2:th.19">FUNCT_2:19</a></i>;<br/>
<b>hence </b><a NAME="E6:30_2"/>
<font color="Maroon" title="c13">GF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">ii</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c6">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">ii</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">C</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">ii</font>
 <b>by </b><i><a class="txt" href="functor0.html#E5:30"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="functor0.html#E1:30_2"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="pboole.html#D24" title="PBOOLE:def.24">PBOOLE:def 24</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:30"/>
<font color="Maroon" title="c10">G</font> <a href="pboole.html#K12" title="PBOOLE:func.12">**</a> <font color="Maroon" title="c9">F</font> is    <a href="functor0.html#M1" title="FUNCTOR0:mode.1">MSUnTrans</a> of <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c8">C</font>
 <b>by </b><i><a class="ref" href="functor0.html#D5" target="_self" title="FUNCTOR0:def.5">Def5</a></i>;<br/>


</div>
