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

<b>set </b><font color="Maroon">F</font> = <font color="Maroon">F<sub>3</sub></font>();<br/>
<b>thus </b><a NAME="E1:16_1"/>
the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>() is <a href="funct_1.html#V2">one-to-one</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D7">FUNCTOR0:def 7</a>,<a class="ref" href="functor0.html#D34">FUNCTOR0:def 34</a>,<a class="ref" href="functor0.html#D36">FUNCTOR0:def 36</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">x2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">y1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">y2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(); <i><font color="Red">:: according to </font></i><a class="ref" href="yellow18.html#D1" target="_self">YELLOW18:def 1</a><br/>

<b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">x1</font>, <font color="Maroon">a2</font> = <font color="Maroon">x2</font>, <font color="Maroon">b1</font> = <font color="Maroon">y1</font>, <font color="Maroon">b2</font> = <font color="Maroon">y2</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<b>assume </b><a NAME="E2:16_1_1"/>
the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="binop_1.html#K2">.</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="binop_1.html#K2">.</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font>
 ;<br/>

<b>then </b><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">a1</font></span>)</span></span><a href="tarski.html#K4">]</a></span> = 
the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="binop_1.html#K2">.</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font>
<b>by </b><i><a class="ref" href="functor0.html#T24">FUNCTOR0:24</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">b2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">a2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="functor0.html#T24">FUNCTOR0:24</a></i>
;<br/>
<a NAME="E4:16_1_1"/><b>then </b>
( <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">b2</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E5:16_1_1"/><b>then </b>
( <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">a1</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">b1</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">b2</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E6:16_1_1"/><b>then </b>
( <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">a1</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">a2</font>) &amp; <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">b1</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">b2</font>) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:16_1_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> )
 <b>by </b><i/>;<br/>


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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2"><b>now </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="E1:16_1_2"/>
<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">F<sub>1</sub></font>(),the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:16_1_2"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() &amp; <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">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font>, <font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i/>;<br/><a NAME="E5:16_1_2"/><i><font color="Green">E49</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>  implies <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">b</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span></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="ref" href="functor0.html#D20">FUNCTOR0:def 20</a></i>;<br/><a NAME="E6:16_1_2"/>
 <a href="functor0.html#K7">Morph-Map</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">a</font>,<font color="Maroon">b</font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:16_1_2_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">x</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#K7">Morph-Map</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> &amp; <font color="Maroon">y</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#K7">Morph-Map</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">x</font>, <font color="Maroon">g</font> = <font color="Maroon">y</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font> ;<br/>
<a NAME="E3:16_1_2_1"/>
( <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K8">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K7">Morph-Map</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K8">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K7">Morph-Map</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, , <a class="ref" href="functor0.html#D17">FUNCTOR0:def 17</a></i>;<br/>
<a NAME="E4:16_1_2_1"/><b>then </b>
( <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</font>) <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K7">Morph-Map</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">g</font>) <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K7">Morph-Map</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:16_1_2_1"/>
(  not <span class="p1">(<span class="default"><a href="functor0.html#K7">Morph-Map</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K7">Morph-Map</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E7:16_1_2"/><b>then </b>
the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> is  <a href="funct_1.html#V2">one-to-one</a> <a href="funct_1.html#NM1">Function</a>
 ;<br/><b>hence </b><a NAME="E8:16_1_2"/>
the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:16_1"/>
the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">F<sub>3</sub></font>() is <a href="msualg_3.html#V1">"1-1"</a>
 <b>by </b><i><a class="ref" href="msualg_3.html#T1">MSUALG_3:1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D31">FUNCTOR0:def 31</a><br/>

<b>reconsider </b><font color="Maroon">G</font> = the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">F<sub>3</sub></font>() as    <a href="pboole.html#M3">ManySortedFunction</a> of the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>(),the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>() <b>by </b><i><a class="ref" href="functor0.html#D5">FUNCTOR0:def 5</a></i>;<br/>
<b>thus </b><a NAME="E5:16_1"/>
<font color="Maroon">F<sub>3</sub></font>() is <a href="functor0.html#V18">full</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D35">FUNCTOR0:def 35</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">G</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D33">FUNCTOR0:def 33</a><br/>

<b>thus </b><a NAME="E1:16_1_3"/>
<font color="Maroon">G</font> <a href="pboole.html#R6">=</a> the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">F<sub>3</sub></font>()
 ;<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="E2:16_1_3"/>
<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">F<sub>1</sub></font>(),the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">ab</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</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">F<sub>1</sub></font>(),the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/>
<a NAME="E4:16_1_3"/>
<font color="Maroon">G</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">ab</font> is   <a href="funct_2.html#NM1">Function</a> of the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">ab</font>,<span class="p1">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ab</font>
 
;<br/>
<b>hence </b><a NAME="E5:16_1_3"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:16_1_3"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">ab</font> <a href="hidden.html#R1">=</a> <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> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font>, <font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i/>;<br/>
the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">ab</font> = 
the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">b</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="functor0.html#T24">FUNCTOR0:24</a></i>
;<br/>
<a NAME="E10:16_1_3"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ab</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">b</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E11:16_1_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">x</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">c</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">d</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">g</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c</font>,<font color="Maroon">d</font><b> such that </b><br/><a NAME="E14:16_1_3"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c</font>) &amp; <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<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>  &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c</font>,<font color="Maroon">d</font>,<font color="Maroon">g</font>) )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E15:16_1_3"/>
( <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">a</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c</font>) &amp; <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">b</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">d</font>) )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E16:16_1_3"/><b>then </b><i><font color="Green">E58</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> )
 
<b>by </b><i/>;<br/>
<i><font color="Green">E62</font></i>: <font color="Maroon">f</font> = 
<font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K8">.</a> <font color="Maroon">g</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="functor0.html#K7">Morph-Map</a> <font color="Maroon">F<sub>3</sub></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="ref" href="functor0.html#D17">FUNCTOR0:def 17</a></i>
;<br/>
<a NAME="E18:16_1_3"/>
<font color="Maroon">G</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">ab</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<a NAME="E19:16_1_3"/><b>then </b>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="functor0.html#K7">Morph-Map</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <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> &amp; <font color="Maroon">G</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">ab</font> <a href="hidden.html#R1">=</a>  <a href="functor0.html#K7">Morph-Map</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">a</font>,<font color="Maroon">b</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E20:16_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><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="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


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

<b>thus </b><a NAME="E6:16_1"/>
 <a href="funcop_1.html#K9">rng</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="tarski.html#R1">c=</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">F<sub>2</sub></font>(),the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></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>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E7:16_1"/>
<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">F<sub>2</sub></font>(),the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:16_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <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> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font>, <font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>2</sub></font>() <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">c</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">c'</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">g</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c</font>,<font color="Maroon">c'</font><b> such that </b><br/><a NAME="E12:16_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c</font>) &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c'</font>) &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">c'</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>  &amp;  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c</font>,<font color="Maroon">c'</font>,<font color="Maroon">g</font>) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">d</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">d'</font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">h</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">d</font>,<font color="Maroon">d'</font><b> such that </b><br/><a NAME="E14:16_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">d</font>) &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">d'</font>) &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">d</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>  &amp;  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">d</font>,<font color="Maroon">d'</font>,<font color="Maroon">h</font>) )
 <b>by </b><i/>;<br/>
<a NAME="E15:16_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">c</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">F<sub>1</sub></font>(),the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E16:16_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>()
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> = 
the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="binop_1.html#K2">.</a> <font color="Maroon">d</font>,<font color="Maroon">c</font>

<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3">.</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></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="functor0.html#T24">FUNCTOR0:24</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></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>, </i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i>, , </i>
;<br/>
<b>hence </b><a NAME="E18:16_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F<sub>3</sub></font>()
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div>
