<?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#L1">AltGraph</a> , <font color="Maroon">B</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="altcat_1.html#L1">AltGraph</a> ;<br/><b>let </b><font color="Maroon">F</font> be  <a href="functor0.html#V10">Covariant</a>  <a href="functor0.html#L2">FunctorStr</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/>



<b>given </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><a NAME="E1:45"/><i><font color="Green">E25</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> &amp; <font color="Maroon">f</font> is <a href="msualg_3.html#V2">"onto"</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D33">FUNCTOR0:def 33</a>,<a class="ref" href="functor0.html#D35">FUNCTOR0:def 35</a><br/>

<b>assume </b><a NAME="E2:45"/><i><font color="Green">E32</font></i>: 
 <a href="funcop_1.html#K9">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>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D8">FUNCTOR0:def 8</a>,<a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a><br/>

<a NAME="E3:45"/><i><font color="Green">E33</font></i>: 
 <a href="relset_1.html#K4">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/>
<b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">B</font>, <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">B</font>;<br/>

<b>assume </b><a NAME="E4:45"/><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>

<a NAME="E5:45"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="funcop_1.html#K9">rng</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:45"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">x</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> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></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#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:45"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="yellow20.html#E3:45"><i><font color="Green">E33</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c</font>, <font color="Maroon">d</font> = <font color="Maroon">d</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font> <b>by </b><i><a class="txt" href="yellow20.html#E4:45"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c</font>
;<br/>

<b>take </b>
<font color="Maroon">d</font>
;<br/>

<a NAME="E11:45"/>
the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font> is <a href="functor0.html#V1">Covariant</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D13">FUNCTOR0:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</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><b> such that </b><br/><a NAME="E13:45"/><i><font color="Green">E52</font></i>: 
the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font> <a href="funct_2.html#R2">=</a> <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K16">:]</a></span>
 <b>by </b><i><a class="ref" href="functor0.html#D2">FUNCTOR0:def 2</a></i>;<br/>
<a NAME="E14:45"/>
( the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d</font>,<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i><a class="ref" href="yellow20.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_3.html#T96">FUNCT_3:96</a></i>;<br/>
<a NAME="E15:45"/><b>then </b>
( <font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a>  )
 
;<br/>
<a NAME="E16:45"/><b>then </b>
( <font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font> &amp; <font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">d</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E17:45"/><b>then </b><i><font color="Green">E53</font></i>: 
the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">d</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="yellow20.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_3.html#T96">FUNCT_3:96</a></i>;<br/>
<a NAME="E18:45"/><b>then </b><i><font color="Green">E54</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">c</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">d</font> )
 
<b>by </b><i><a class="txt" href="yellow20.html#E3:45"><i><font color="Green">E33</font></i></a>, <a class="txt" href="yellow20.html#E4:45"><i><font color="Green">E34</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="functor0.html#K5">Morph-Map</a> <font color="Maroon">F</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></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"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="txt" href="yellow20.html#E3:45"><i><font color="Green">E33</font></i></a>, <a class="txt" href="yellow20.html#E4:45"><i><font color="Green">E34</font></i></a>, <a class="ref" href="msualg_3.html#D3">MSUALG_3:def 3</a></i>
<br/>.= 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
<b>by </b><i><a class="txt" href="yellow20.html#E3:45"><i><font color="Green">E33</font></i></a>, <a class="txt" href="yellow20.html#E4:45"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/>
<b>then consider </b><font color="Maroon">g</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E21:45"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="functor0.html#K5">Morph-Map</a> <font color="Maroon">F</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K5">Morph-Map</a> <font color="Maroon">F</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> )
 <b>by </b><i><a class="txt" href="yellow20.html#E2:45"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c</font>,<font color="Maroon">d</font> <b>by </b><i><a class="ref" href="yellow20.html#T5" target="_self">Th5</a></i>;<br/>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>thus </b><a NAME="E23:45"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">c</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">d</font> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="yellow20.html#E3:45"><i><font color="Green">E33</font></i></a>, <a class="txt" href="yellow20.html#E4:45"><i><font color="Green">E34</font></i></a>, <a class="ref" href="yellow20.html#T4" target="_self">Th4</a>, <a class="ref" href="yellow20.html#T5" target="_self">Th5</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>

<b>thus </b><a NAME="E24:45"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="functor0.html#K6">.</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="yellow20.html#E2:45"><i><font color="Green">E32</font></i></a>, , <a class="ref" href="yellow20.html#T5" target="_self">Th5</a>, <a class="ref" href="functor0.html#D16">FUNCTOR0:def 16</a></i>;<br/>


</div>
