<?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  <a href="cat_1.html#V1">strict</a> <a href="cat_5.html#V1" target="_self">with_triple-like_morphisms</a> <a href="cat_1.html#NM4">Category</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:9_1"/><i><font color="Green">E5</font></i>: 
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E2:9_1"/><i><font color="Green">E6</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:9_1"/><i><font color="Green">E7</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()ex <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>] )
 <b>and </b><br/><a NAME="E4:9_1"/><i><font color="Green">E8</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="cat_1.html#K4">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>6</sub></font>)</span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E5:9_1"/><i><font color="Green">E9</font></i>: 
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E6:9_1"/><i><font color="Green">E10</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E7:9_1"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()ex <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>] )
 <b>and </b><br/><a NAME="E8:9_1"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="cat_1.html#K4">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>6</sub></font>)</span><a href="tarski.html#K4">]</a></span>
 ;<br/>

<a NAME="E9:9_1"/><i><font color="Green">E13</font></i>: 
the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:9_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:9_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()ex <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] )
 <b>by </b><i><a class="txt" href="cat_5.html#E3:9_1"><i><font color="Green">E7</font></i></a></i>;<br/><a NAME="E3:9_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="cat_5.html#E6:9_1"><i><font color="Green">E10</font></i></a></i>;<br/><b>hence </b><a NAME="E4:9_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>3</sub></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="E2:9_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E3:9_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()ex <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] )
 
<b>by </b><i><a class="txt" href="cat_5.html#E7:9_1"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E4:9_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="cat_5.html#E2:9_1"><i><font color="Green">E6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:9_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:9_1"/><i><font color="Green">E14</font></i>: 
(  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:9_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E9:9_1"><i><font color="Green">E13</font></i></a></i>;<br/><b>thus </b>the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> = 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>4</sub></font>

<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="cat_5.html#K4" target="_self">`11</a> 
<b>by </b><i><a class="ref" href="cat_5.html#T2" target="_self">Th2</a></i>
<br/>.= 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="cat_5.html#T2" target="_self">Th2</a></i>
<br/>.= 
the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>

;<br/></div><b>end;</b></div>
<a NAME="E12:9_1"/><b>then </b><i><font color="Green">E15</font></i>: 
the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="cat_5.html#E9:9_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="cat_5.html#E10:9_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:9_1_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E9:9_1"><i><font color="Green">E13</font></i></a></i>;<br/><b>thus </b>the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> = 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>4</sub></font>

<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="cat_5.html#K5" target="_self">`12</a> 
<b>by </b><i><a class="ref" href="cat_5.html#T2" target="_self">Th2</a></i>
<br/>.= 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="cat_5.html#T2" target="_self">Th2</a></i>
<br/>.= 
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>

;<br/></div><b>end;</b></div>
<a NAME="E14:9_1"/><b>then </b><i><font color="Green">E16</font></i>: 
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="cat_5.html#E9:9_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="cat_5.html#E10:9_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:9_1_4"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E4:9_1_4"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>]
 <b>and </b><br/><a NAME="E5:9_1_4"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() holds <br/>( ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] implies <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>) <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) &amp; ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>2</sub></font>] implies <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>2</sub></font>) <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) )
 <b>by </b><i><a class="txt" href="cat_5.html#E1:9"><i><font color="Green">E4</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E1:9_1"><i><font color="Green">E5</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">c<sub>9</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E8:9_1_4"/><i><font color="Green">E19</font></i>: 
(  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>] )
 <b>by </b><i><a class="txt" href="cat_5.html#E3:9_1"><i><font color="Green">E7</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E5:9_1"><i><font color="Green">E9</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">c<sub>13</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E11:9_1_4"/><i><font color="Green">E20</font></i>: 
(  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>] )
 <b>by </b><i><a class="txt" href="cat_5.html#E7:9_1"><i><font color="Green">E11</font></i></a></i>;<br/><a NAME="E12:9_1_4"/>
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>;<br/><a NAME="E13:9_1_4"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="cat_5.html#K4" target="_self">`11</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="cat_5.html#K5" target="_self">`12</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="cat_5.html#K4" target="_self">`11</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="cat_5.html#K5" target="_self">`12</a>  )
 <b>by </b><i><a class="ref" href="cat_5.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E14:9_1_4"/><b>then </b><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E8:9_1_4"><i><font color="Green">E19</font></i></a>, <a class="txt" href="cat_5.html#E11:9_1_4"><i><font color="Green">E20</font></i></a>, <a class="ref" href="mcart_1.html#T89">MCART_1:89</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E2:9_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="cat_5.html#E4:9_1_4"><i><font color="Green">E17</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E6:9_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="cat_5.html#E4:9_1_4"><i><font color="Green">E17</font></i></a></i>;<br/> <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>14</sub></font> = 
<font color="Maroon">c<sub>14</sub></font> <a href="cat_5.html#K5" target="_self">`12</a> 
<b>by </b><i><a class="ref" href="cat_5.html#T2" target="_self">Th2</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#T89">MCART_1:89</a></i>
;<br/><b>then </b><i><font color="Green">E22</font></i>: <font color="Maroon">c<sub>14</sub></font> = 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="cat_1.html#T46">CAT_1:46</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font>)</span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E4:9_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="cat_5.html#E8:9_1_4"><i><font color="Green">E19</font></i></a>, <a class="txt" href="cat_5.html#E14:9_1_4"><i><font color="Green">E21</font></i></a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E5:9_1_4"><i><font color="Green">E18</font></i></a>, <a class="txt" href="cat_5.html#E8:9_1_4"><i><font color="Green">E19</font></i></a>, <a class="txt" href="cat_5.html#E14:9_1_4"><i><font color="Green">E21</font></i></a></i>
;<br/> <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>15</sub></font> = 
<font color="Maroon">c<sub>15</sub></font> <a href="cat_5.html#K5" target="_self">`12</a> 
<b>by </b><i><a class="ref" href="cat_5.html#T2" target="_self">Th2</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#T89">MCART_1:89</a></i>
;<br/><b>then </b><i><font color="Green">E23</font></i>: <font color="Maroon">c<sub>15</sub></font> = 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="ref" href="cat_1.html#T46">CAT_1:46</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>5</sub></font>)</span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E8:9_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="cat_5.html#E11:9_1_4"><i><font color="Green">E20</font></i></a>, <a class="txt" href="cat_5.html#E14:9_1_4"><i><font color="Green">E21</font></i></a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E5:9_1_4"><i><font color="Green">E18</font></i></a>, <a class="txt" href="cat_5.html#E11:9_1_4"><i><font color="Green">E20</font></i></a>, <a class="txt" href="cat_5.html#E14:9_1_4"><i><font color="Green">E21</font></i></a></i>
;<br/><b>thus </b>the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> = 
 <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>6</sub></font>

<br/>.= 
the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="cat_5.html#E8:9_1_4"><i><font color="Green">E19</font></i></a>, <a class="txt" href="cat_5.html#E11:9_1_4"><i><font color="Green">E20</font></i></a>, <a class="txt" href="cat_5.html#E14:9_1_4"><i><font color="Green">E21</font></i></a>, <a class="txt" href="cat_5.html#E18:9_1_4"><i><font color="Green">E22</font></i></a>, <a class="txt" href="cat_5.html#E20:9_1_4"><i><font color="Green">E23</font></i></a></i>
;<br/></div><b>end;</b></div>
<a NAME="E16:9_1"/><b>then </b><i><font color="Green">E17</font></i>: 
the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#R1">=</a> the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="cat_5.html#E1:9_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="cat_5.html#E5:9_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="cat_5.html#E10:9_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E17:9_1"/><i><font color="Green">E18</font></i>: 
(  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</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="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>2</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="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E9:9_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E18:9_1"/><i><font color="Green">E19</font></i>: 
 <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_5"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:9_1_5_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></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="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></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="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="cat_1.html#U2">Morphisms</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="cat_5.html#E9:9_1"><i><font color="Green">E13</font></i></a></i>;<br/><a NAME="E4:9_1_5_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_5.html#K3" target="_self">`2</a> </span>)</span></span><a href="cat_2.html#K13">]</a></span>
 <b>by </b><i><a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/><a NAME="E5:9_1_5_1"/><b>then </b>
the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cat_5.html#K3" target="_self">`2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="cat_5.html#E1:9_1_5_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E6:9_1_5_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="cat_5.html#E9:9_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="cat_5.html#E12:9_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="cat_5.html#E14:9_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="cat_5.html#E4:9_1_5_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>3</sub></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="E2:9_1_5"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></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="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i><a class="txt" href="cat_5.html#E17:9_1"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></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="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="cat_1.html#U2">Morphisms</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="cat_5.html#E9:9_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E5:9_1_5"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="cat_5.html#K3" target="_self">`2</a> </span>)</span></span><a href="cat_2.html#K13">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<a NAME="E6:9_1_5"/><b>then </b>
the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="domain_1.html#K2">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="cat_5.html#K3" target="_self">`2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="cat_5.html#E2:9_1_5"><i><font color="Green">E20</font></i></a>, <a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E7:9_1_5"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="cat_5.html#E9:9_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="cat_5.html#E12:9_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="cat_5.html#E14:9_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="cat_5.html#E5:9_1_5"><i><font color="Green">E21</font></i></a>, <a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:9_1_6"/><i><font color="Green">E20</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></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="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E9:9_1"><i><font color="Green">E13</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">c<sub>11</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E5:9_1_6"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>] )
 <b>by </b><i><a class="txt" href="cat_5.html#E3:9_1"><i><font color="Green">E7</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">c<sub>14</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E7:9_1_6"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>] )
 <b>by </b><i><a class="txt" href="cat_5.html#E3:9_1"><i><font color="Green">E7</font></i></a></i>;<br/><i><font color="Green">E23</font></i>: <font color="Maroon">c<sub>9</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="cat_5.html#K4" target="_self">`11</a> 
<b>by </b><i><a class="txt" href="cat_5.html#E5:9_1_6"><i><font color="Green">E21</font></i></a>, <a class="ref" href="mcart_1.html#T89">MCART_1:89</a></i>
<br/>.= 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="cat_5.html#T2" target="_self">Th2</a></i>
<br/>.= 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="cat_5.html#E1:9_1_6"><i><font color="Green">E20</font></i></a>, <a class="ref" href="cat_1.html#T40">CAT_1:40</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="cat_5.html#K5" target="_self">`12</a> 
<b>by </b><i><a class="ref" href="cat_5.html#T2" target="_self">Th2</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="cat_5.html#E7:9_1_6"><i><font color="Green">E22</font></i></a>, <a class="ref" href="mcart_1.html#T89">MCART_1:89</a></i>
;<br/><b>thus </b>the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> = 
<font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="cat_5.html#E1:9_1_6"><i><font color="Green">E20</font></i></a>, <a class="ref" href="cat_1.html#D4">CAT_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>14</sub></font>)</span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E4:9_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="cat_5.html#E5:9_1_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="cat_5.html#E7:9_1_6"><i><font color="Green">E22</font></i></a>, <a class="txt" href="cat_5.html#E8:9_1_6"><i><font color="Green">E23</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="cat_5.html#E8:9_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="cat_5.html#E5:9_1_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="cat_5.html#E7:9_1_6"><i><font color="Green">E22</font></i></a>, <a class="txt" href="cat_5.html#E8:9_1_6"><i><font color="Green">E23</font></i></a></i>
<br/>.= 
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E18:9_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="cat_5.html#E1:9_1_6"><i><font color="Green">E20</font></i></a>, <a class="ref" href="cat_1.html#D4">CAT_1:def 4</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E20:9_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="cat_5.html#E1:9_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="cat_5.html#E5:9_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="cat_5.html#E9:9_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="cat_5.html#E12:9_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="cat_5.html#E14:9_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="cat_5.html#E16:9_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="cat_5.html#E18:9_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="partfun1.html#T35">PARTFUN1:35</a></i>;<br/>


</div>
