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

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#R2">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Olive">b<sub>2</sub></font> <a href="group_6.html#NR2" target="_self">is_epimorphism</a>  ) </span>}</span> ;<br/>
<a NAME="E1:3_1_1"/><i><font color="Green">E3</font></i>: 
 <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#R2">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Olive">b<sub>2</sub></font> <a href="group_6.html#NR2" target="_self">is_epimorphism</a>  ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:3_1_1_1"/><i><font color="Green">E4</font></i>: 
 <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> as   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="group_6.html#T47">GROUP_6:47</a></i>;<br/>
<a NAME="E3:3_1_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/>
<a NAME="E4:3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="group_6.html#NR2" target="_self">is_epimorphism</a> 
 
<b>by </b><i><a class="ref" href="group_6.html#D13">GROUP_6:def 13</a></i>;<br/>
<b>hence </b><a NAME="E5:3_1_1_1"/>
 <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#R2">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Olive">b<sub>2</sub></font> <a href="group_6.html#NR2" target="_self">is_epimorphism</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="autgroup.html#E1:3_1_1_1"><i><font color="Green">E4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#R2">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Olive">b<sub>2</sub></font> <a href="group_6.html#NR2" target="_self">is_epimorphism</a>  ) </span>}</span>  as    <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E3:3_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="fraenkel.html#V1">functional</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="fraenkel.html#D1">FRAENKEL:def 1</a><br/>

<b>assume </b><a NAME="E1:3_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:3_1_1_2"/><i><font color="Green">E4</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="group_6.html#NR2" target="_self">is_epimorphism</a>  ) )
 ;<br/>
<b>thus </b><a NAME="E4:3_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> is   <a href="funct_1.html#NM1">Function</a>
 <b>by </b><i><a class="txt" href="autgroup.html#E3:3_1_1_2"><i><font color="Green">E4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="fraenkel.html#V1">functional</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="autgroup.html#E1:3_1_1"><i><font color="Green">E3</font></i></a></i>;<br/>
<a NAME="E5:3_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is    <a href="fraenkel.html#M1">FUNCTION_DOMAIN</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="fraenkel.html#D2">FRAENKEL:def 2</a><br/>

<a NAME="E1:3_1_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:3_1_1_3"/><i><font color="Green">E4</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="group_6.html#NR2" target="_self">is_epimorphism</a>  ) )
 ;<br/>
<b>thus </b><a NAME="E4:3_1_1_3"/>
<font color="Maroon">c<sub>5</sub></font> is   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="autgroup.html#E3:3_1_1_3"><i><font color="Green">E4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="fraenkel.html#M1">FUNCTION_DOMAIN</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="fraenkel.html#M2">Element</a> of <font color="Maroon">c<sub>5</sub></font>;<br/><a NAME="E1:3_1_1_4"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:3_1_1_4"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#R2">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>2</sub></font> being  <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#R2">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Olive">b<sub>2</sub></font> <a href="group_6.html#NR2" target="_self">is_epimorphism</a>  ) )
 ;<br/><b>hence </b><a NAME="E3:3_1_1_4"/>
<font color="Maroon">c<sub>6</sub></font> is   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:3_1_1_5"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:3_1_1_5"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; ex <font color="Olive">b<sub>2</sub></font> being  <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#R2">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Olive">b<sub>2</sub></font> <a href="group_6.html#NR2" target="_self">is_epimorphism</a>  ) )
 ;<br/><b>hence </b><a NAME="E3:3_1_1_5"/>
( <font color="Maroon">c<sub>6</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="group_6.html#NR2" target="_self">is_epimorphism</a>  )
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E9:3_1_1"/><i><font color="Green">E4</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="group_6.html#NR2" target="_self">is_epimorphism</a>  )
 ;<br/>

<a NAME="E10:3_1_1"/>
<font color="Maroon">c<sub>6</sub></font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>hence </b><a NAME="E11:3_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="autgroup.html#E9:3_1_1"><i><font color="Green">E4</font></i></a></i>;<br/>


</div>
