<?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="setfam_1.html#V1">with_non-empty_elements</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="E1:24"/><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="E2:24"/><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="24_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1"/>
( 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>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="realset1.html#V1">trivial</a> or ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> is <a href="realset1.html#V1">trivial</a> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1"><b>suppose </b></a><a NAME="E1:24_1_1"/><i><font color="Green">E21</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>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="realset1.html#V1">trivial</a>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:24_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="altcat_3.html#V5" target="_self">epi</a> implies <font color="Maroon">c<sub>5</sub></font> is <a href="funct_2.html#V2">onto</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="altcat_3.html#V5" target="_self">epi</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_1_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_1_1_1_1_1"/><i><font color="Green">E22</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><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_1_1_1_1_1_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="24_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_1_1_1_1_1_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="E2:24_1_1_1_1_1_1_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="24_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_1_1_1_1_1_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="E2:24_1_1_1_1_1_1_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#E1:24_1_1_1_1_1_1"><i><font color="Green">E22</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>hence </b><a NAME="E3:24_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="funct_2.html#V2">onto</a>
 <b>by </b><i><a class="txt" href="altcat_3.html#E1:24_1_1_1_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_1_1_1_1_2"/><i><font color="Green">E22</font></i>: 
<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="E2:24_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="altcat_1.html#D16">ALTCAT_1:def 16</a></i>;<br/><a NAME="E3:24_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i><a class="txt" href="altcat_3.html#E1:24_1_1"><i><font color="Green">E21</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:24_1_1_1_1_1_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="altcat_3.html#E1:24_1_1_1_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="realset1.html#D4">REALSET1:def 4</a></i>;<br/><a NAME="E6:24_1_1_1_1_1_2"/><i><font color="Green">E24</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="altcat_3.html#E5:24_1_1_1_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><a NAME="E7:24_1_1_1_1_1_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="altcat_1.html#D16">ALTCAT_1:def 16</a></i>;<br/><a NAME="E8:24_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E26</font></i>: 
<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/><a NAME="E9:24_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i><a class="txt" href="altcat_3.html#E1:24_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="altcat_3.html#E7:24_1_1_1_1_1_2"><i><font color="Green">E25</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:24_1_1_1_1_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="altcat_3.html#E8:24_1_1_1_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="realset1.html#D4">REALSET1:def 4</a></i>;<br/><a NAME="E12:24_1_1_1_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="altcat_3.html#E1:24_1_1_1_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="altcat_3.html#E11:24_1_1_1_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E13:24_1_1_1_1_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <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="relat_1.html#T65">RELAT_1:65</a></i>;<br/><a NAME="E14:24_1_1_1_1_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="altcat_3.html#E6:24_1_1_1_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/><b>hence </b><a NAME="E15:24_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="funct_2.html#V2">onto</a>
 <b>by </b><i><a class="txt" href="altcat_3.html#E5:24_1_1_1_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:24_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="funct_2.html#V2">onto</a>
 ;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:24_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> is <a href="funct_2.html#V2">onto</a> implies <font color="Maroon">c<sub>4</sub></font> is <a href="altcat_3.html#V5" target="_self">epi</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1_1_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="funct_2.html#V2">onto</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#D8" target="_self">ALTCAT_3:def 8</a><br/>

<b>assume </b><a NAME="E2:24_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>3</sub></font>,<font color="Maroon">c<sub>6</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>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/>

<b>assume </b><a NAME="E3:24_1_1_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E4:24_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>3</sub></font>,<font color="Maroon">c<sub>6</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>3</sub></font>,<font color="Maroon">c<sub>6</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="E6:24_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>3</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>6</sub></font> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E2:24_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="E8:24_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>3</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>6</sub></font> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E2:24_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="altcat_3.html#E4:24_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="E9:24_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>2</sub></font>,<font color="Maroon">c<sub>6</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#E1:24"><i><font color="Green">E19</font></i></a>, <a class="txt" href="altcat_3.html#E2:24_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>9</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>8</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="altcat_3.html#E1:24"><i><font color="Green">E19</font></i></a>, <a class="txt" href="altcat_3.html#E2:24"><i><font color="Green">E20</font></i></a>, <a class="txt" href="altcat_3.html#E2:24_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="altcat_3.html#E3:24_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="altcat_3.html#E6:24_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>10</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="altcat_3.html#E1:24"><i><font color="Green">E19</font></i></a>, <a class="txt" href="altcat_3.html#E2:24"><i><font color="Green">E20</font></i></a>, <a class="txt" href="altcat_3.html#E2:24_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="altcat_3.html#E8:24_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="altcat_3.html#E9:24_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="24_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_1_2_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:24_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>3</sub></font> &amp; <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#NR2" target="_self">&lt;&gt;</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#E6:24_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="altcat_3.html#E8:24_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/><a NAME="E4:24_1_1_2_2"/>
<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>5</sub></font>
 <b>by </b><i><a class="txt" href="altcat_3.html#E1:24_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="altcat_3.html#E3:24_1_1_2_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:24_1_1_2_2"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>12</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>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:24_1_1_2_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="altcat_3.html#E6:24_1_1_2_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><b>hence </b><a NAME="E8:24_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="altcat_3.html#E10:24_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="altcat_3.html#E3:24_1_1_2_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="altcat_3.html#E6:24_1_1_2_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:24_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#E6:24_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="altcat_3.html#E8:24_1_1_2"><i><font color="Green">E27</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2"><b>suppose </b></a><a NAME="E1:24_1_2"/><i><font color="Green">E21</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> is <a href="realset1.html#V1">trivial</a> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_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="E1:24_1_2_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>  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="24_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_2_1_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="E2:24_1_2_1_1_1"/><b>then </b><i><font color="Green">E22</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>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:24_1_2_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> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E5:24_1_2_1_1_1"/><i><font color="Green">E24</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i><a class="txt" href="altcat_3.html#E1:24_1_2"><i><font color="Green">E21</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <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> <b>by </b><i><a class="txt" href="altcat_3.html#E4:24_1_2_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="altcat_1.html#D16">ALTCAT_1:def 16</a></i>;<br/><b>thus </b><a NAME="E7:24_1_2_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="altcat_3.html#V5" target="_self">epi</a> implies <font color="Maroon">c<sub>5</sub></font> is <a href="funct_2.html#V2">onto</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:24_1_2_1_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="altcat_3.html#V5" target="_self">epi</a>
 <b>and </b><br/><a NAME="E2:24_1_2_1_1_1_1"/><i><font color="Green">E26</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> is <a href="funct_2.html#V2">onto</a>
 ;<br/>

<a NAME="E3:24_1_2_1_1_1_1"/><i><font color="Green">E27</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E4:24_1_2_1_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E2:24_1_2_1_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E5:24_1_2_1_1_1_1"/><b>then </b>
not <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</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#E3:24_1_2_1_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:24_1_2_1_1_1_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:24_1_2_1_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <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#E4:24_1_2_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="setfam_1.html#D9">SETFAM_1:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="altcat_3.html#E5:24_1_2_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="altcat_3.html#E8:24_1_2_1_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="realset1.html#T4">REALSET1:4</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>10</sub></font>;<br/>
<a NAME="E12:24_1_2_1_1_1_1"/><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>7</sub></font> &amp; not <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E14:24_1_2_1_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E12:24_1_2_1_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E15:24_1_2_1_1_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="altcat_3.html#E8:24_1_2_1_1_1_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>9</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E17:24_1_2_1_1_1_1"/><i><font color="Green">E33</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E18:24_1_2_1_1_1_1"/><i><font color="Green">E34</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>3</sub></font> holds <br/><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<b>set </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>9</sub></font>;<br/>
<a NAME="E19:24_1_2_1_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>3</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E20:24_1_2_1_1_1_1"/><i><font color="Green">E36</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>3</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>9</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>9</sub></font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> = 
 <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="altcat_3.html#E7:24_1_2_1_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="altcat_3.html#E18:24_1_2_1_1_1_1"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/>
<a NAME="E22:24_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
not <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E7:24_1_2_1_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="altcat_3.html#E14:24_1_2_1_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="altcat_3.html#E20:24_1_2_1_1_1_1"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E23:24_1_2_1_1_1_1"/><i><font color="Green">E38</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</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="E1:24_1_2_1_1_1_1_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:24_1_2_1_1_1_1_2"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_2_1_1_1_1_2_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> or <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:24_1_2_1_1_1_1_2_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><font color="Maroon">c<sub>15</sub></font> = 
 <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="altcat_3.html#E17:24_1_2_1_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="altcat_3.html#E18:24_1_2_1_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="altcat_3.html#E3:24_1_2_1_1_1_1_2"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="altcat_3.html#E1:24_1_2_1_1_1_1_2_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/><b>hence </b><a NAME="E3:24_1_2_1_1_1_1_2_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="altcat_3.html#E8:24_1_2_1_1_1_1"><i><font color="Green">E29</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:24_1_2_1_1_1_1_2_1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><font color="Maroon">c<sub>15</sub></font> = 
 <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="altcat_3.html#E17:24_1_2_1_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="altcat_3.html#E18:24_1_2_1_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="altcat_3.html#E3:24_1_2_1_1_1_1_2"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="altcat_3.html#E1:24_1_2_1_1_1_1_2_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/><b>hence </b><a NAME="E3:24_1_2_1_1_1_1_2_1_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="altcat_3.html#E8:24_1_2_1_1_1_1"><i><font color="Green">E29</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E24:24_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E17:24_1_2_1_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E25:24_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D16">ALTCAT_1:def 16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="altcat_3.html#E24:24_1_2_1_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="altcat_1.html#D16">ALTCAT_1:def 16</a></i>;<br/>
<a NAME="E27:24_1_2_1_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</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="E1:24_1_2_1_1_1_1_3"/>
<font color="Maroon">c<sub>16</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>3</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>17</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:24_1_2_1_1_1_1_3"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>17</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>3</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</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="E4:24_1_2_1_1_1_1_3"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="altcat_3.html#E15:24_1_2_1_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="altcat_3.html#E19:24_1_2_1_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="altcat_3.html#E20:24_1_2_1_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="altcat_3.html#E3:24_1_2_1_1_1_1_3"><i><font color="Green">E41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E28:24_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E19:24_1_2_1_1_1_1"><i><font color="Green">E35</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>16</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>9</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="ref" href="altcat_1.html#D16">ALTCAT_1:def 16</a></i>;<br/>
<a NAME="E30:24_1_2_1_1_1_1"/><i><font color="Green">E41</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></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#E2:24_1_2_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="altcat_3.html#E3:24_1_2_1_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="altcat_3.html#E17:24_1_2_1_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/>
<a NAME="E31:24_1_2_1_1_1_1"/>
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>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T25">FUNCT_1:25</a></i>;<br/>
<a NAME="E32:24_1_2_1_1_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E33:24_1_2_1_1_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E23:24_1_2_1_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E34:24_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>5</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>7</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E30:24_1_2_1_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E35:24_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <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>7</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D16">ALTCAT_1:def 16</a></i>;<br/>
<a NAME="E36:24_1_2_1_1_1_1"/><i><font color="Green">E43</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></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#E2:24_1_2_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="altcat_3.html#E3:24_1_2_1_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="altcat_3.html#E19:24_1_2_1_1_1_1"><i><font color="Green">E35</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="24_1_2_1_1_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:24_1_2_1_1_1_1_4"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:24_1_2_1_1_1_1_4"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</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#E2:24_1_2_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then </b><i><font color="Green">E46</font></i>: <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> = 
 <a href="cqc_lang.html#K1">IFEQ</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="altcat_3.html#E3:24_1_2_1_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="altcat_3.html#E18:24_1_2_1_1_1_1"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="altcat_3.html#E7:24_1_2_1_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="altcat_3.html#E2:24_1_2_1_1_1_1_4"><i><font color="Green">E45</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> = 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_3.html#E30:24_1_2_1_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="altcat_3.html#E1:24_1_2_1_1_1_1_4"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_3.html#E3:24_1_2_1_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="altcat_3.html#E20:24_1_2_1_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="altcat_3.html#E2:24_1_2_1_1_1_1_4"><i><font color="Green">E45</font></i></a>, <a class="txt" href="altcat_3.html#E3:24_1_2_1_1_1_1_4"><i><font color="Green">E46</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="altcat_3.html#E36:24_1_2_1_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="altcat_3.html#E1:24_1_2_1_1_1_1_4"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E38:24_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E30:24_1_2_1_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="altcat_3.html#E36:24_1_2_1_1_1_1"><i><font color="Green">E43</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>15</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="altcat_3.html#E1:24"><i><font color="Green">E19</font></i></a>, <a class="txt" href="altcat_3.html#E2:24"><i><font color="Green">E20</font></i></a>, <a class="txt" href="altcat_3.html#E25:24_1_2_1_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="altcat_3.html#E35:24_1_2_1_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="altcat_1.html#T18">ALTCAT_1:18</a></i>
<br/>.= 
<font color="Maroon">c<sub>16</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="altcat_3.html#E1:24"><i><font color="Green">E19</font></i></a>, <a class="txt" href="altcat_3.html#E2:24"><i><font color="Green">E20</font></i></a>, <a class="txt" href="altcat_3.html#E25:24_1_2_1_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="altcat_3.html#E35:24_1_2_1_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="altcat_1.html#T18">ALTCAT_1:18</a></i>
;<br/>
<b>hence </b><a NAME="E40:24_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="altcat_3.html#E1:24_1_2_1_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="altcat_3.html#E22:24_1_2_1_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="altcat_3.html#E25:24_1_2_1_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="altcat_3.html#D8" target="_self">Def8</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E8:24_1_2_1_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> is <a href="funct_2.html#V2">onto</a> implies <font color="Maroon">c<sub>4</sub></font> is <a href="altcat_3.html#V5" target="_self">epi</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1_2_1_1_1_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="funct_2.html#V2">onto</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>8</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#D8" target="_self">ALTCAT_3:def 8</a><br/>

<b>assume </b><a NAME="E2:24_1_2_1_1_1_2"/><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>3</sub></font>,<font color="Maroon">c<sub>8</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>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font>;<br/>

<b>assume </b><a NAME="E3:24_1_2_1_1_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E4:24_1_2_1_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>3</sub></font>,<font color="Maroon">c<sub>8</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>3</sub></font>,<font color="Maroon">c<sub>8</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>11</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:24_1_2_1_1_1_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E2:24_1_2_1_1_1_2"><i><font color="Green">E26</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>12</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:24_1_2_1_1_1_2"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</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>3</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>8</sub></font> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E2:24_1_2_1_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="altcat_3.html#E4:24_1_2_1_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E9:24_1_2_1_1_1_2"/><i><font color="Green">E31</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>8</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#E1:24"><i><font color="Green">E19</font></i></a>, <a class="txt" href="altcat_3.html#E2:24_1_2_1_1_1_2"><i><font color="Green">E26</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">E32</font></i>: <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>10</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="altcat_3.html#E1:24"><i><font color="Green">E19</font></i></a>, <a class="txt" href="altcat_3.html#E2:24"><i><font color="Green">E20</font></i></a>, <a class="txt" href="altcat_3.html#E2:24_1_2_1_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="altcat_3.html#E3:24_1_2_1_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="altcat_3.html#E6:24_1_2_1_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="altcat_1.html#T18">ALTCAT_1:18</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="altcat_3.html#E1:24"><i><font color="Green">E19</font></i></a>, <a class="txt" href="altcat_3.html#E2:24"><i><font color="Green">E20</font></i></a>, <a class="txt" href="altcat_3.html#E2:24_1_2_1_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="altcat_3.html#E8:24_1_2_1_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="altcat_3.html#E9:24_1_2_1_1_1_2"><i><font color="Green">E31</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="24_1_2_1_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_2_1_1_1_2_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font>
 ;<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="E3:24_1_2_1_1_1_2_2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E6:24_1_2_1_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="altcat_3.html#E8:24_1_2_1_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><a NAME="E4:24_1_2_1_1_1_2_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</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#E1:24_1_2_1_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="altcat_3.html#E3:24_1_2_1_1_1_2_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<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="E6:24_1_2_1_1_1_2_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> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <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/><a NAME="E7:24_1_2_1_1_1_2_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></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#E6:24_1_2_1_1_1_2_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><b>hence </b><a NAME="E8:24_1_2_1_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="altcat_3.html#E10:24_1_2_1_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="altcat_3.html#E3:24_1_2_1_1_1_2_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="altcat_3.html#E6:24_1_2_1_1_1_2_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:24_1_2_1_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="altcat_3.html#E6:24_1_2_1_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="altcat_3.html#E8:24_1_2_1_1_1_2"><i><font color="Green">E30</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_2_1_1_2"/><i><font color="Green">E22</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><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_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="E1:24_1_2_1_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="24_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_2_1_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="E2:24_1_2_1_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="24_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_2_1_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="E2:24_1_2_1_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#E1:24_1_2_1_1_2"><i><font color="Green">E22</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>hence </b><a NAME="E3:24_1_2_1_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="altcat_3.html#V5" target="_self">epi</a> implies <font color="Maroon">c<sub>5</sub></font> is <a href="funct_2.html#V2">onto</a> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E1:24_1_2_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/><b>thus </b><a NAME="E4:24_1_2_1_1_2"/>
( <font color="Maroon">c<sub>5</sub></font> is <a href="funct_2.html#V2">onto</a> implies <font color="Maroon">c<sub>4</sub></font> is <a href="altcat_3.html#V5" target="_self">epi</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1_2_1_1_2_2"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="funct_2.html#V2">onto</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#D8" target="_self">ALTCAT_3:def 8</a><br/>

<b>assume </b><a NAME="E2:24_1_2_1_1_2_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>3</sub></font>,<font color="Maroon">c<sub>6</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>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/>

<b>assume </b><a NAME="E3:24_1_2_1_1_2_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E4:24_1_2_1_1_2_2"/><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>3</sub></font>,<font color="Maroon">c<sub>6</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>3</sub></font>,<font color="Maroon">c<sub>6</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="E6:24_1_2_1_1_2_2"/><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>3</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>6</sub></font> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E2:24_1_2_1_1_2_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="E8:24_1_2_1_1_2_2"/><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>3</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>6</sub></font> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E2:24_1_2_1_1_2_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="altcat_3.html#E4:24_1_2_1_1_2_2"><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="E9:24_1_2_1_1_2_2"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>10</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#E1:24_1_2_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="altcat_3.html#E6:24_1_2_1_1_2_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="altcat_3.html#E8:24_1_2_1_1_2_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/>
<b>hence </b><a NAME="E10:24_1_2_1_1_2_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#E6:24_1_2_1_1_2_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="altcat_3.html#E8:24_1_2_1_1_2_2"><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><b>end;</b></div><b>hence </b><a NAME="E3:24_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="altcat_3.html#V5" target="_self">epi</a> iff <font color="Maroon">c<sub>5</sub></font> is <a href="funct_2.html#V2">onto</a> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
