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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></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_1.html#L2">AltCatStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="functor0.html#V15" target="_self">covariant</a>  <a href="functor0.html#M2" target="_self">Functor</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><a NAME="E1:121"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="functor0.html#V21" target="_self">bijective</a>
 ;<br/>

<a NAME="E2:121"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="functor0.html#V19" target="_self">injective</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D36" target="_self">Def36</a></i>;<br/>
<a NAME="E3:121"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="functor0.html#V4" target="_self">one-to-one</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D34" target="_self">Def34</a></i>;<br/>
<a NAME="E4:121"/><b>then </b><i><font color="Green">E81</font></i>: 
the <a href="functor0.html#U1" target="_self">ObjectMap</a> of <font color="Maroon">c<sub>3</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E5:121"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="functor0.html#V8" target="_self">feasible</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D26" target="_self">Def26</a></i>;<br/>
<a NAME="E6:121"/><b>then </b><i><font color="Green">E83</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="functor0.html#K15" target="_self">"</a>  is <a href="functor0.html#V21" target="_self">bijective</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="functor0.html#K15" target="_self">"</a>  is <a href="functor0.html#V8" target="_self">feasible</a> )
 
<b>by </b><i><a class="txt" href="functor0.html#E1:121"><i><font color="Green">E80</font></i></a>, <a class="ref" href="functor0.html#T36" target="_self">Th36</a></i>;<br/>
<a NAME="E7:121"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="functor0.html#V12" target="_self">id-preserving</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D26" target="_self">Def26</a></i>;<br/>
<a NAME="E8:121"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="functor0.html#V13" target="_self">comp-preserving</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D27" target="_self">Def27</a></i>;<br/>
<a NAME="E9:121"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="functor0.html#V20" target="_self">surjective</a>
 
<b>by </b><i><a class="txt" href="functor0.html#E1:121"><i><font color="Green">E80</font></i></a>, <a class="ref" href="functor0.html#D36" target="_self">Def36</a></i>;<br/>
<a NAME="E10:121"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="functor0.html#V5" target="_self">onto</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D35" target="_self">Def35</a></i>;<br/>
<a NAME="E11:121"/><b>then </b><i><font color="Green">E87</font></i>: 
the <a href="functor0.html#U1" target="_self">ObjectMap</a> of <font color="Maroon">c<sub>3</sub></font> is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E12:121"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="functor0.html#V10" target="_self">Covariant</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D27" target="_self">Def27</a></i>;<br/>
<a NAME="E13:121"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="functor0.html#V7" target="_self">coreflexive</a>
 
<b>by </b><i><a class="txt" href="functor0.html#E10:121"><i><font color="Green">E86</font></i></a>, <a class="ref" href="functor0.html#T47" target="_self">Th47</a></i>;<br/>
<a NAME="E14:121"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="functor0.html#K15" target="_self">"</a>  is <a href="functor0.html#V10" target="_self">Covariant</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_1"><b>proof </b></a><div class="add">

<a NAME="E1:121_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="functor0.html#V10" target="_self">Covariant</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D27" target="_self">Def27</a></i>;<br/>
<a NAME="E2:121_1"/><b>then </b>
the <a href="functor0.html#U1" target="_self">ObjectMap</a> of <font color="Maroon">c<sub>3</sub></font> is <a href="functor0.html#V1" target="_self">Covariant</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D13" target="_self">Def13</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:121_1"/><i><font color="Green">E91</font></i>: 
the <a href="functor0.html#U1" target="_self">ObjectMap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="funct_3.html#K16">:]</a></span>
 <b>by </b><i><a class="ref" href="functor0.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E5:121_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="functor0.html#E4:121"><i><font color="Green">E81</font></i></a>, <a class="txt" href="functor0.html#E4:121_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="functor0.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E6:121_1"/><b>then </b><i><font color="Green">E93</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<a NAME="E7:121_1"/><i><font color="Green">E94</font></i>: 
 <a href="funcop_1.html#K9">rng</a> <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="funct_3.html#K16">:]</a></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">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="functor0.html#E11:121"><i><font color="Green">E87</font></i></a>, <a class="txt" href="functor0.html#E4:121_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E8:121_1"/>
 <a href="funcop_1.html#K9">rng</a> <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="funct_3.html#K16">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_3.html#T88">FUNCT_3:88</a></i>;<br/>
<a NAME="E9:121_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="functor0.html#E7:121_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="zfmisc_1.html#T115">ZFMISC_1:115</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K2">"</a>  as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="functor0.html#E6:121_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D2" target="_self">FUNCTOR0:def 2</a>,<a class="ref" href="functor0.html#D13" target="_self">FUNCTOR0:def 13</a><br/>

<a NAME="E11:121_1"/>
<span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="funct_3.html#K16">:]</a></span> <a href="funct_1.html#K2">"</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="funct_3.html#K16">:]</a></span>
 
<b>by </b><i><a class="txt" href="functor0.html#E5:121_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="functor0.html#T7" target="_self">Th7</a></i>;<br/>
<b>hence </b><a NAME="E12:121_1"/>
the <a href="functor0.html#U1" target="_self">ObjectMap</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="functor0.html#K15" target="_self">"</a> </span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="funct_3.html#K16">:]</a></span>
 <b>by </b><i><a class="txt" href="functor0.html#E1:121"><i><font color="Green">E80</font></i></a>, <a class="txt" href="functor0.html#E4:121_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="functor0.html#D39" target="_self">Def39</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:121"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="functor0.html#K15" target="_self">"</a>  is <a href="functor0.html#V12" target="_self">id-preserving</a>
 
<b>by </b><i><a class="txt" href="functor0.html#E1:121"><i><font color="Green">E80</font></i></a>, <a class="txt" href="functor0.html#E5:121"><i><font color="Green">E82</font></i></a>, <a class="txt" href="functor0.html#E7:121"><i><font color="Green">E84</font></i></a>, <a class="txt" href="functor0.html#E13:121"><i><font color="Green">E89</font></i></a>, <a class="ref" href="functor0.html#T38" target="_self">Th38</a></i>;<br/>
<a NAME="E16:121"/>
<font color="Maroon">c<sub>3</sub></font> <a href="functor0.html#K15" target="_self">"</a>  is <a href="functor0.html#V13" target="_self">comp-preserving</a>
 
<b>by </b><i><a class="txt" href="functor0.html#E1:121"><i><font color="Green">E80</font></i></a>, <a class="txt" href="functor0.html#E5:121"><i><font color="Green">E82</font></i></a>, <a class="txt" href="functor0.html#E8:121"><i><font color="Green">E85</font></i></a>, <a class="txt" href="functor0.html#E12:121"><i><font color="Green">E88</font></i></a>, <a class="txt" href="functor0.html#E13:121"><i><font color="Green">E89</font></i></a>, <a class="ref" href="functor0.html#T43" target="_self">Th43</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="functor0.html#K15" target="_self">"</a>  as    <a href="functor0.html#M2" target="_self">Functor</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="functor0.html#E6:121"><i><font color="Green">E83</font></i></a>, <a class="txt" href="functor0.html#E14:121"><i><font color="Green">E90</font></i></a>, <a class="txt" href="functor0.html#E15:121"><i><font color="Green">E91</font></i></a>, <a class="ref" href="functor0.html#D26" target="_self">Def26</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>thus </b><a NAME="E18:121"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="functor0.html#K15" target="_self">"</a> 
 ;<br/>

<b>thus </b><a NAME="E19:121"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="functor0.html#V21" target="_self">bijective</a>
 <b>by </b><i><a class="txt" href="functor0.html#E1:121"><i><font color="Green">E80</font></i></a>, <a class="txt" href="functor0.html#E5:121"><i><font color="Green">E82</font></i></a>, <a class="ref" href="functor0.html#T36" target="_self">Th36</a></i>;<br/>

<b>thus </b><a NAME="E20:121"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="functor0.html#V10" target="_self">Covariant</a>
 <b>by </b><i><a class="txt" href="functor0.html#E14:121"><i><font color="Green">E90</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D27" target="_self">FUNCTOR0:def 27</a><br/>

<b>thus </b><a NAME="E21:121"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="functor0.html#V13" target="_self">comp-preserving</a>
 <b>by </b><i><a class="txt" href="functor0.html#E1:121"><i><font color="Green">E80</font></i></a>, <a class="txt" href="functor0.html#E5:121"><i><font color="Green">E82</font></i></a>, <a class="txt" href="functor0.html#E8:121"><i><font color="Green">E85</font></i></a>, <a class="txt" href="functor0.html#E12:121"><i><font color="Green">E88</font></i></a>, <a class="txt" href="functor0.html#E13:121"><i><font color="Green">E89</font></i></a>, <a class="ref" href="functor0.html#T43" target="_self">Th43</a></i>;<br/>


</div>
