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

<b>set </b><font color="Maroon">CC</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>;<br/>
<a NAME="E1:91_1_1"/><i><font color="Green">E20</font></i>: 
the <a href="functor0.html#U1" target="_self">ObjectMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">A</font></span>)</span> <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>
 
<b>by </b><i><a class="ref" href="functor0.html#T6" target="_self">Th6</a></i>;<br/>
<b>thus </b><a NAME="E2:91_1_1"/>
 <a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">A</font> is <a href="functor0.html#V4" target="_self">one-to-one</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D34" target="_self">FUNCTOR0:def 34</a>,<a class="ref" href="functor0.html#D36" target="_self">FUNCTOR0:def 36</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:91_1_1_1"/>
the <a href="functor0.html#U1" target="_self">ObjectMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">A</font></span>)</span> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="functor0.html#T11" target="_self">Th11</a>, <a class="ref" href="funct_1.html#T52">FUNCT_1:52</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D7" target="_self">FUNCTOR0:def 7</a><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:91_1_1"/>
 <a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">A</font> is <a href="functor0.html#V17" target="_self">faithful</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_1_1_2_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <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="91_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:91_1_1_2_1_1"/><i><font color="Green">E23</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><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="msualg_3.html#D2">MSUALG_3:def 2</a>,<a class="ref" href="functor0.html#D31" target="_self">FUNCTOR0:def 31</a><br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E2:91_1_1_2_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="functor0.html#U2" target="_self">MorphMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">A</font></span>)</span>
 <b>and </b><br/><a NAME="E3:91_1_1_2_1_1"/><i><font color="Green">E30</font></i>: 
the <a href="functor0.html#U2" target="_self">MorphMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E4:91_1_1_2_1_1"/>
 <a href="relat_1.html#K1">dom</a> the <a href="functor0.html#U2" target="_self">MorphMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">A</font></span>)</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">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="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">o1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">o2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E6:91_1_1_2_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , <a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/><b>reconsider </b><font color="Maroon">o1</font> = <font color="Maroon">o1</font>, <font color="Maroon">o2</font> = <font color="Maroon">o2</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font> ;<br/><a NAME="E8:91_1_1_2_1_1"/><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="tarski.html#K4">]</a></span> <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>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><font color="Maroon">f</font> = 
<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="binop_1.html#K1">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>
<b>by </b><i>, , <a class="ref" href="functor0.html#T6" target="_self">Th6</a></i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="msualg_3.html#D1">MSUALG_3:def 1</a></i>
;<br/><b>hence </b><a NAME="E10:91_1_1_2_1_1"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#T52">FUNCT_1:52</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:91_1_1_2_1_2"/><i><font color="Green">E34</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><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="msualg_3.html#D2">MSUALG_3:def 2</a>,<a class="ref" href="functor0.html#D31" target="_self">FUNCTOR0:def 31</a><br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E2:91_1_1_2_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="functor0.html#U2" target="_self">MorphMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">A</font></span>)</span>
 <b>and </b><br/><a NAME="E3:91_1_1_2_1_2"/>
the <a href="functor0.html#U2" target="_self">MorphMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 ;<br/> <a href="relat_1.html#K1">dom</a> the <a href="functor0.html#U2" target="_self">MorphMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">A</font></span>)</span> = 
<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="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>
;<br/><b>hence </b><a NAME="E5:91_1_1_2_1_2"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>thus </b><a NAME="E4:91_1_1"/>
 <a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">A</font> is <a href="functor0.html#V18" target="_self">full</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D35" target="_self">FUNCTOR0:def 35</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_1_1_3_1"/>
(  not <font color="Maroon">A</font> is <a href="struct_0.html#V3">empty</a> or <font color="Maroon">A</font> is <a href="struct_0.html#V3">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:91_1_1_3_1_1"/>
not <font color="Maroon">A</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as  non <a href="struct_0.html#V3">empty</a>  <a href="altcat_1.html#L1">AltGraph</a>  ;<br/><a NAME="E3:91_1_1_3_1_1"/>
 <a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">A</font> is <a href="functor0.html#V18" target="_self">full</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_3_1_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">f</font> = the <a href="functor0.html#U2" target="_self">MorphMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">A</font></span>)</span> as    <a href="pboole.html#M3">ManySortedFunction</a> of 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> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1" target="_self">ObjectMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D33" target="_self">FUNCTOR0:def 33</a><br/>

<b>thus </b><a NAME="E2:91_1_1_3_1_1_1"/>
<font color="Maroon">f</font> <a href="pboole.html#R6">=</a> the <a href="functor0.html#U2" target="_self">MorphMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">A</font></span>)</span>
 ;<br/>

<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="msualg_3.html#D3">MSUALG_3:def 3</a><br/>

<b>assume </b><a NAME="E3:91_1_1_3_1_1_1"/><i><font color="Green">E37</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 consider </b><font color="Maroon">o1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">o2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E5:91_1_1_3_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">o1</font> = <font color="Maroon">o1</font>, <font color="Maroon">o2</font> = <font color="Maroon">o2</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font> ;<br/>
<a NAME="E7:91_1_1_3_1_1_1"/><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="tarski.html#K4">]</a></span> <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>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E8:91_1_1_3_1_1_1"/><i><font color="Green">E47</font></i>: 
 <a href="relset_1.html#K4">dom</a> the <a href="functor0.html#U1" target="_self">ObjectMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">A</font></span>)</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">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#T11" target="_self">Th11</a>, <a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<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="binop_1.html#K1">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>
<b>by </b><i>, <a class="ref" href="functor0.html#T6" target="_self">Th6</a></i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="msualg_3.html#D1">MSUALG_3:def 1</a></i>
;<br/>
<b>hence </b> <a href="relat_1.html#K2">rng</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> = 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</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> <span class="p1">(<span class="default">the <a href="functor0.html#U1" target="_self">ObjectMap</a> of <span class="p2">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="functor0.html#T11" target="_self">Th11</a>, , , <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1" target="_self">ObjectMap</a> of <span class="p2">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:91_1_1_3_1_1"/>
 <a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">A</font> is <a href="functor0.html#V18" target="_self">full</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:91_1_1_3_1_2"/>
<font color="Maroon">A</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><div class="add"><a NAME="E2:91_1_1_3_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
the <a href="struct_0.html#U1">carrier</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="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E3:91_1_1_3_1_2"/>
the <a href="functor0.html#U1" target="_self">ObjectMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">A</font></span>)</span> <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>
 <b>by </b><i><a class="ref" href="functor0.html#T6" target="_self">Th6</a></i>;<br/><b>then reconsider </b><font color="Maroon">B</font> = the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> <a href="funct_1.html#NK3" target="_self">*</a> the <a href="functor0.html#U1" target="_self">ObjectMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">A</font></span>)</span> as    <a href="pboole.html#M1">ManySortedSet</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> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">f</font> = the <a href="functor0.html#U2" target="_self">MorphMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">A</font></span>)</span> as    <a href="pboole.html#M1">ManySortedSet</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> ;<br/><a NAME="E6:91_1_1_3_1_2"/>
<font color="Maroon">f</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">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_3_1_2_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>thus </b><a NAME="E1:91_1_1_3_1_2_1"/>
(  not <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> or <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is    <a href="relset_1.html#M2">Relation</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>,<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="txt" href="functor0.html#E2:91_1_1_3_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <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">B</font> ;<br/><b>take </b>
<font color="Maroon">B</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D32" target="_self">FUNCTOR0:def 32</a><br/><b>take </b>
<font color="Maroon">f</font>
;<br/><b>thus </b><a NAME="E8:91_1_1_3_1_2"/>
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> <a href="funct_1.html#NK3" target="_self">*</a> the <a href="functor0.html#U1" target="_self">ObjectMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">A</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="pboole.html#R6">=</a> the <a href="functor0.html#U2" target="_self">MorphMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">A</font></span>)</span> )
 ;<br/><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="msualg_3.html#D3">MSUALG_3:def 3</a><br/><b>thus </b><a NAME="E9:91_1_1_3_1_2"/>
(  not <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> or  <a href="relat_1.html#K2">rng</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> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="txt" href="functor0.html#E2:91_1_1_3_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>thus </b><a NAME="E5:91_1_1"/>
 <a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">A</font> is <a href="functor0.html#V5" target="_self">onto</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:91_1_1_4"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2"><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></span>)</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">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="relat_1.html#T71">RELAT_1:71</a></i>;<br/>
<b>hence </b><a NAME="E2:91_1_1_4"/>
the <a href="functor0.html#U1" target="_self">ObjectMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">A</font></span>)</span> is <a href="funct_2.html#V2">onto</a>
 <b>by </b><i><a class="ref" href="functor0.html#T11" target="_self">Th11</a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D8" target="_self">FUNCTOR0:def 8</a><br/>


</div><b>end;</b></div>


</div>
