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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:23"/><i><font color="Green">E19</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>



<b>assume </b><a NAME="E2:23"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1"><b>suppose </b></a><a NAME="E1:23_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:23_1_1"/><b>then </b><i><font color="Green">E21</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <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/><b>thus </b><a NAME="E3:23_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="altcat_3.html#V4" target="_self">mono</a> implies <font color="Maroon">c<sub>5</sub></font> is <a href="funct_1.html#V2">one-to-one</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="altcat_3.html#V4" target="_self">mono</a>
 ;<br/>

<b>assume </b><a NAME="E2:23_1_1_1"/>
not <font color="Maroon">c<sub>5</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:23_1_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>and </b><br/><a NAME="E5:23_1_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E6:23_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E7:23_1_1_1"/><i><font color="Green">E26</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></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="ref" href="altcat_1.html#T23">ALTCAT_1:23</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>6</sub></font>;<br/>
<a NAME="E8:23_1_1_1"/><i><font color="Green">E27</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E9:23_1_1_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>7</sub></font>;<br/>
<a NAME="E10:23_1_1_1"/><i><font color="Green">E29</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E11:23_1_1_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E13:23_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E2:23_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="altcat_3.html#E4:23_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="altcat_3.html#E9:23_1_1_1"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E14:23_1_1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E2:23_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="altcat_3.html#E4:23_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="altcat_3.html#E6:23_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="altcat_3.html#E11:23_1_1_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E15:23_1_1_1"/><i><font color="Green">E32</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</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="E1:23_1_1_1_1"/>
<font color="Maroon">c<sub>12</sub></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">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:23_1_1_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>thus </b><a NAME="E4:23_1_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="altcat_3.html#E2:23_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="altcat_3.html#E4:23_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="altcat_3.html#E8:23_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="altcat_3.html#E9:23_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="altcat_3.html#E3:23_1_1_1_1"><i><font color="Green">E33</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:23_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E8:23_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>6</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="altcat_1.html#D16">ALTCAT_1:def 16</a></i>;<br/>
<a NAME="E18:23_1_1_1"/><i><font color="Green">E33</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</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="E1:23_1_1_1_2"/>
<font color="Maroon">c<sub>13</sub></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">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:23_1_1_1_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>thus </b><a NAME="E4:23_1_1_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="altcat_3.html#E2:23_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="altcat_3.html#E4:23_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="altcat_3.html#E10:23_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="altcat_3.html#E11:23_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="altcat_3.html#E3:23_1_1_1_2"><i><font color="Green">E34</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:23_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E10:23_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>7</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="altcat_1.html#D16">ALTCAT_1:def 16</a></i>;<br/>
<a NAME="E21:23_1_1_1"/><i><font color="Green">E34</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E2:23_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="altcat_3.html#E8:23_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="altcat_3.html#E15:23_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E22:23_1_1_1"/><i><font color="Green">E35</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E2:23_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="altcat_3.html#E10:23_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="altcat_3.html#E18:23_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:23_1_1_1_3"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_3.html#E21:23_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="altcat_3.html#E5:23_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="altcat_3.html#E9:23_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="altcat_3.html#E1:23_1_1_1_3"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_3.html#E11:23_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="altcat_3.html#E1:23_1_1_1_3"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="altcat_3.html#E22:23_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="altcat_3.html#E1:23_1_1_1_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
;<br/><br/></div><b>end;</b></div>
<a NAME="E24:23_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E21:23_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="altcat_3.html#E22:23_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>4</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>12</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_3.html#E1:23"><i><font color="Green">E19</font></i></a>, <a class="txt" href="altcat_3.html#E2:23"><i><font color="Green">E20</font></i></a>, <a class="txt" href="altcat_3.html#E7:23_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="altcat_1.html#T18">ALTCAT_1:18</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="altcat_3.html#E1:23"><i><font color="Green">E19</font></i></a>, <a class="txt" href="altcat_3.html#E2:23"><i><font color="Green">E20</font></i></a>, <a class="txt" href="altcat_3.html#E7:23_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="altcat_1.html#T18">ALTCAT_1:18</a></i>
;<br/>
<b>hence </b><a NAME="E26:23_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="altcat_3.html#E1:23_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="altcat_3.html#E7:23_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="altcat_3.html#E14:23_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="altcat_3.html#D7" target="_self">Def7</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E4:23_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> is <a href="funct_1.html#V2">one-to-one</a> implies <font color="Maroon">c<sub>4</sub></font> is <a href="altcat_3.html#V4" target="_self">mono</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1_1_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D7" target="_self">ALTCAT_3:def 7</a><br/>

<b>assume </b><a NAME="E2:23_1_1_2"/><i><font color="Green">E23</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E3:23_1_1_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E4:23_1_1_2"/><i><font color="Green">E25</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D16">ALTCAT_1:def 16</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:23_1_1_2"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E2:23_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:23_1_1_2"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E2:23_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="altcat_3.html#E4:23_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E9:23_1_1_2"/><i><font color="Green">E28</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></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="altcat_3.html#E1:23"><i><font color="Green">E19</font></i></a>, <a class="txt" href="altcat_3.html#E2:23_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<br/>
<b>then </b><i><font color="Green">E29</font></i>: <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="altcat_3.html#E1:23"><i><font color="Green">E19</font></i></a>, <a class="txt" href="altcat_3.html#E2:23"><i><font color="Green">E20</font></i></a>, <a class="txt" href="altcat_3.html#E2:23_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="altcat_3.html#E3:23_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="altcat_3.html#E6:23_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="altcat_1.html#T18">ALTCAT_1:18</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="altcat_3.html#E1:23"><i><font color="Green">E19</font></i></a>, <a class="txt" href="altcat_3.html#E2:23"><i><font color="Green">E20</font></i></a>, <a class="txt" href="altcat_3.html#E2:23_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="altcat_3.html#E8:23_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="altcat_3.html#E9:23_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="altcat_1.html#T18">ALTCAT_1:18</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:23_1_1_2_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:23_1_1_2_2"/><b>then </b><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E6:23_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="altcat_3.html#E8:23_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:23_1_1_2_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="altcat_3.html#E6:23_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="altcat_3.html#E1:23_1_1_2_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E4:23_1_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="altcat_3.html#E8:23_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="altcat_3.html#E10:23_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="altcat_3.html#E1:23_1_1_2_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><b>hence </b><a NAME="E5:23_1_1_2_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="altcat_3.html#E2:23_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="altcat_3.html#E1:23_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="altcat_3.html#E6:23_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="altcat_3.html#E8:23_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="altcat_3.html#E2:23_1_1_2_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:23_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="altcat_3.html#E6:23_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="altcat_3.html#E8:23_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2"><b>suppose </b></a><a NAME="E1:23_1_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><div><i><font color="Green">E22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_2_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:23_1_2_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:23_1_2_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="partfun1.html#T57">PARTFUN1:57</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:23_1_2_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:23_1_2_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="altcat_3.html#E1:23_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>thus </b><a NAME="E3:23_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="altcat_3.html#V4" target="_self">mono</a> implies <font color="Maroon">c<sub>5</sub></font> is <a href="funct_1.html#V2">one-to-one</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1_2_2"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="altcat_3.html#V4" target="_self">mono</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="xboole_0.html#K1">{}</a> , <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="altcat_3.html#E2:23_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_2.html#T55">FUNCT_2:55</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>
<a NAME="E3:23_1_2_2"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="partfun1.html#T59">PARTFUN1:59</a></i>;<br/>
<b>hence </b><a NAME="E4:23_1_2_2"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E4:23_1_2"/>
( <font color="Maroon">c<sub>5</sub></font> is <a href="funct_1.html#V2">one-to-one</a> implies <font color="Maroon">c<sub>4</sub></font> is <a href="altcat_3.html#V4" target="_self">mono</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1_2_3"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7">EnsCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D7" target="_self">ALTCAT_3:def 7</a><br/>

<b>assume </b><a NAME="E2:23_1_2_3"/><i><font color="Green">E23</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E3:23_1_2_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E4:23_1_2_3"/><i><font color="Green">E24</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D16">ALTCAT_1:def 16</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:23_1_2_3"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E2:23_1_2_3"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:23_1_2_3"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E2:23_1_2_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="altcat_3.html#E4:23_1_2_3"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E9:23_1_2_3"/><i><font color="Green">E27</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D16">ALTCAT_1:def 16</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E12:23_1_2_3"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E1:23"><i><font color="Green">E19</font></i></a>, <a class="txt" href="altcat_3.html#E9:23_1_2_3"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E13:23_1_2_3"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="altcat_3.html#E1:23_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="altcat_3.html#E12:23_1_2_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E14:23_1_2_3"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/>
<a NAME="E15:23_1_2_3"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="altcat_3.html#E6:23_1_2_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="altcat_3.html#E12:23_1_2_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E16:23_1_2_3"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/>
<font color="Maroon">c<sub>9</sub></font> = 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="altcat_3.html#E15:23_1_2_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="altcat_3.html#E6:23_1_2_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="altcat_3.html#E8:23_1_2_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="altcat_3.html#E16:23_1_2_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>
;<br/>
<b>hence </b><a NAME="E18:23_1_2_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="altcat_3.html#E6:23_1_2_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="altcat_3.html#E8:23_1_2_3"><i><font color="Green">E26</font></i></a></i>;<br/>


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

</div>
