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

<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_2.html#V1">reflexive</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_2.html#V1">reflexive</a>  <a href="altcat_1.html#L2">AltCatStr</a> ;<br/><b>let </b><font color="Maroon">F</font> be  <a href="functor0.html#V8">feasible</a>  <a href="functor0.html#L2">FunctorStr</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/>



<b>assume </b><a NAME="E1:22"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">F</font> is <a href="functor0.html#V21">bijective</a>
 ;<br/>

<b>let </b><font color="Maroon">G</font> be  <a href="functor0.html#V8">feasible</a>  <a href="functor0.html#L2">FunctorStr</a> of <font color="Maroon">B</font>,<font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E2:22"/><i><font color="Green">E31</font></i>: 
 <a href="functor0.html#G2">FunctorStr</a>(# the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font>,the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">G</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="functor0.html#K15">"</a> 
 ;<br/>

<b>set </b><font color="Maroon">I1</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/>
<b>set </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">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>;<br/>
<a NAME="E3:22"/>
the <a href="functor0.html#U1">ObjectMap</a> of <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K13">*</a> <font color="Maroon">G</font></span>)</span> <a href="funct_2.html#R2">=</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="functor0.html#D37">FUNCTOR0:def 37</a></i>;<br/>
<a NAME="E4:22"/><b>then </b><i><font color="Green">E34</font></i>: 
the <a href="functor0.html#U1">ObjectMap</a> of <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K13">*</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default">the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font> <a href="funct_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i>, , <a class="ref" href="functor0.html#D39">FUNCTOR0:def 39</a></i>;<br/>
<a NAME="E5:22"/>
<font color="Maroon">F</font> is <a href="functor0.html#V19">injective</a>
 
<b>by </b><i>, <a class="ref" href="functor0.html#D36">FUNCTOR0:def 36</a></i>;<br/>
<a NAME="E6:22"/><b>then </b>
<font color="Maroon">F</font> is <a href="functor0.html#V4">one-to-one</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D34">FUNCTOR0:def 34</a></i>;<br/>
<a NAME="E7:22"/><b>then </b><i><font color="Green">E35</font></i>: 
the <a href="functor0.html#U1">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="ref" href="functor0.html#D7">FUNCTOR0:def 7</a></i>;<br/>
<a NAME="E8:22"/>
<font color="Maroon">F</font> is <a href="functor0.html#V20">surjective</a>
 
<b>by </b><i>, <a class="ref" href="functor0.html#D36">FUNCTOR0:def 36</a></i>;<br/>
<a NAME="E9:22"/><b>then </b>
<font color="Maroon">F</font> is <a href="functor0.html#V5">onto</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D35">FUNCTOR0:def 35</a></i>;<br/>
<a NAME="E10:22"/><b>then </b><i><font color="Green">E36</font></i>: 
the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font> is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D8">FUNCTOR0:def 8</a></i>;<br/>
<b>set </b><font color="Maroon">OM</font> = the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font>;<br/>
<a NAME="E11:22"/><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font> <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="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:22"/><i><font color="Green">E38</font></i>: 
 <a href="relat_1.html#K2">rng</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font> <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/>
<b>reconsider </b><font color="Maroon">OM</font> = the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font> 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">B</font> ;<br/>
<a NAME="E14:22"/><i><font color="Green">E43</font></i>: 
the <a href="functor0.html#U1">ObjectMap</a> of <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K13">*</a> <font color="Maroon">G</font></span>)</span> <a href="funct_2.html#R2">=</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">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_1.html#T61">FUNCT_1:61</a></i>;<br/>
<a NAME="E15:22"/>
<font color="Maroon">F</font> is <a href="functor0.html#V19">injective</a>
 
<b>by </b><i>, <a class="ref" href="functor0.html#D36">FUNCTOR0:def 36</a></i>;<br/>
<a NAME="E16:22"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">F</font> is <a href="functor0.html#V17">faithful</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D34">FUNCTOR0:def 34</a></i>;<br/>
<a NAME="E17:22"/><b>then </b><i><font color="Green">E45</font></i>: 
the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">F</font> is <a href="msualg_3.html#V1">"1-1"</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D31">FUNCTOR0:def 31</a></i>;<br/>
<a NAME="E18:22"/>
<font color="Maroon">F</font> is <a href="functor0.html#V20">surjective</a>
 
<b>by </b><i>, <a class="ref" href="functor0.html#D36">FUNCTOR0:def 36</a></i>;<br/>
<a NAME="E19:22"/><b>then </b><i><font color="Green">E46</font></i>: 
( <font color="Maroon">F</font> is <a href="functor0.html#V18">full</a> &amp; <font color="Maroon">F</font> is <a href="functor0.html#V5">onto</a> )
 
<b>by </b><i><a class="ref" href="functor0.html#D35">FUNCTOR0:def 35</a></i>;<br/>
<a NAME="E20:22"/>
 <a href="functor0.html#G2">FunctorStr</a>(# the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font>,the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">G</font> #) is <a href="functor0.html#V21">bijective</a>
 
<b>by </b><i>, , <a class="ref" href="functor0.html#T36">FUNCTOR0:36</a></i>;<br/>
<a NAME="E21:22"/><b>then </b>
 <a href="functor0.html#G2">FunctorStr</a>(# the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font>,the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">G</font> #) is <a href="functor0.html#V20">surjective</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D36">FUNCTOR0:def 36</a></i>;<br/>
<a NAME="E22:22"/><b>then </b>
(  <a href="functor0.html#G2">FunctorStr</a>(# the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font>,the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">G</font> #) is <a href="functor0.html#V18">full</a> &amp;  <a href="functor0.html#G2">FunctorStr</a>(# the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font>,the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">G</font> #) is <a href="functor0.html#V5">onto</a> )
 
<b>by </b><i><a class="ref" href="functor0.html#D35">FUNCTOR0:def 35</a></i>;<br/>
<a NAME="E23:22"/><b>then </b><i><font color="Green">E49</font></i>: 
the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font> is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D8">FUNCTOR0:def 8</a></i>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <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">B</font> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E25:22"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">k</font> <a href="pboole.html#R6">=</a> the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E26:22"/><i><font color="Green">E76</font></i>: 
the <a href="functor0.html#U2">MorphMap</a> of <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K15">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default">the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font> <a href="funct_1.html#K2">"</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="functor0.html#D39">FUNCTOR0:def 39</a></i>;<br/>
<b>consider </b><font color="Maroon">f</font> being    <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">B</font> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E28:22"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">f</font> <a href="pboole.html#R6">=</a> the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E29:22"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">f</font> is <a href="msualg_3.html#V2">"onto"</a>
 <b>by </b><i>, <a class="ref" href="functor0.html#D33">FUNCTOR0:def 33</a></i>;<br/>
<a NAME="E30:22"/><i><font color="Green">E80</font></i>: 
the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font> <a href="funct_1.html#K2">"</a> 
 
<b>by </b><i>, , <a class="ref" href="functor0.html#D39">FUNCTOR0:def 39</a></i>;<br/>
<b>set </b><font color="Maroon">OMG</font> = the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font>;<br/>
<a NAME="E31:22"/><i><font color="Green">E81</font></i>: 
the <a href="functor0.html#U2">MorphMap</a> of <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K13">*</a> <font color="Maroon">G</font></span>)</span> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font></span>)</span> <a href="altcat_2.html#K3">**</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i>, , , , <a class="ref" href="functor0.html#D37">FUNCTOR0:def 37</a></i>;<br/>
<a NAME="E32:22"/><i><font color="Green">E82</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">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> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font></span>)</span> <a href="altcat_2.html#K3">**</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E33:22"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">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">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> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font></span>)</span> <a href="altcat_2.html#K3">**</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:22_1"/><i><font color="Green">E83</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">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>
 ;<br/>

<a NAME="E2:22_1"/><i><font color="Green">E84</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font> <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#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:22_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="functor1.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E4:22_1"/><i><font color="Green">E86</font></i>: 
the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:22_1_1"/>
 <a href="relat_1.html#K2">rng</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font> <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="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E2:22_1_1"/>
the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <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>
 <b>by </b><i><a class="ref" href="functor1.html#T3" target="_self">Th3</a>, <a class="txt" href="functor1.html#E2:22"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E87</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="functor1.html#T3" target="_self">Th3</a>, <a class="txt" href="functor1.html#E2:22"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> 
<b>by </b><i>, , , , , <a class="ref" href="msualg_3.html#D5">MSUALG_3:def 5</a></i>
;<br/>
<a NAME="E6:22_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:22_1_3"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">f</font> is <a href="msualg_3.html#V1">"1-1"</a>
 
<b>by </b><i>, , <a class="ref" href="functor0.html#D31">FUNCTOR0:def 31</a></i>;<br/>
<a NAME="E2:22_1_3"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <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>hence </b><a NAME="E3:22_1_3"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, <a class="txt" href="functor1.html#E10:22"><i><font color="Green">E36</font></i></a>, <a class="ref" href="msualg_3.html#D2">MSUALG_3:def 2</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E97</font></i>:  <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> <span class="p2">(<span class="default">the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="msualg_3.html#D3">MSUALG_3:def 3</a></i>
<br/>.= 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<i><font color="Green">E98</font></i>: the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">OM</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">OM</font> <a href="funct_1.html#K2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="functor1.html#T3" target="_self">Th3</a>, <a class="txt" href="functor1.html#E2:22"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<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">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></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#T61">FUNCT_1:61</a></i>
<br/>.= 
<font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="functor1.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font></span>)</span> <a href="altcat_2.html#K3">**</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default">the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span>
<b>by </b><i>, , <a class="txt" href="functor1.html#E1:22"><i><font color="Green">E29</font></i></a>, <a class="ref" href="functor1.html#T3" target="_self">Th3</a>, , <a class="txt" href="functor1.html#E4:22"><i><font color="Green">E34</font></i></a>, <a class="ref" href="pboole.html#D24">PBOOLE:def 24</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">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="txt" href="functor1.html#E7:22"><i><font color="Green">E35</font></i></a>, <a class="txt" href="functor1.html#E10:22"><i><font color="Green">E36</font></i></a>, <a class="txt" href="functor1.html#E11:22"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_1.html#T61">FUNCT_1:61</a></i>
<br/>.= 
<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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="functor1.html#T3" target="_self">Th3</a>, <a class="ref" href="msualg_3.html#D1">MSUALG_3:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E10:22_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font></span>)</span> <a href="altcat_2.html#K3">**</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <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">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E34:22"/><b>then </b>
the <a href="functor0.html#U2">MorphMap</a> of <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K13">*</a> <font color="Maroon">G</font></span>)</span> <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">B</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>
<b>hence </b><a NAME="E35:22"/>
<font color="Maroon">F</font> <a href="functor0.html#K13">*</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="functor0.html#K12">id</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="functor0.html#D30">FUNCTOR0:def 30</a></i>;<br/>


</div>
