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<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <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>assume </b><b>that </b><br/><a NAME="E1:3_1_2"/><i><font color="Green">E3</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="fraenkel.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Olive">b<sub>1</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> ) &amp; ( for <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> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> iff ( <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>  ) ) ) )
 <b>and </b><br/><a NAME="E2:3_1_2"/><i><font color="Green">E4</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="fraenkel.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> holds <font color="Olive">b<sub>1</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> ) &amp; ( for <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> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff ( <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/>

<a NAME="E3:3_1_2"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_2_1"><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="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:3_1_2_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</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="txt" href="autgroup.html#E1:3_1_2"><i><font color="Green">E3</font></i></a></i>;<br/>
<a NAME="E3:3_1_2_1"/>
( <font color="Maroon">c<sub>5</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="group_6.html#NR2" target="_self">is_epimorphism</a>  )
 
<b>by </b><i><a class="txt" href="autgroup.html#E1:3_1_2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="autgroup.html#E1:3_1_2_1"><i><font color="Green">E6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:3_1_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="autgroup.html#E2:3_1_2"><i><font color="Green">E4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:3_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <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="3_1_2_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="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:3_1_2_2"/><i><font color="Green">E6</font></i>: 
<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 reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</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="txt" href="autgroup.html#E2:3_1_2"><i><font color="Green">E4</font></i></a></i>;<br/>
<a NAME="E3:3_1_2_2"/>
( <font color="Maroon">c<sub>5</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="group_6.html#NR2" target="_self">is_epimorphism</a>  )
 
<b>by </b><i><a class="txt" href="autgroup.html#E2:3_1_2"><i><font color="Green">E4</font></i></a>, <a class="txt" href="autgroup.html#E1:3_1_2_2"><i><font color="Green">E6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:3_1_2_2"/>
<font color="Maroon">c<sub>4</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="autgroup.html#E1:3_1_2"><i><font color="Green">E3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:3_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="autgroup.html#E3:3_1_2"><i><font color="Green">E5</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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