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<b>let </b><font color="Maroon" title="c2">F1</font>, <font color="Maroon" title="c3">F2</font> be    <a href="fraenkel.html#M1" title="FRAENKEL:mode.1">FUNCTION_DOMAIN</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">f</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of <font color="Maroon" title="c2">F1</font> holds  <font color="Olive" title="b1">f</font> is   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c1">G</font> ) &amp; ( for <font color="Olive" title="b1">h</font> being  <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c1">G</font> holds <br/> ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> iff ( <font color="Olive" title="b1">h</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Olive" title="b1">h</font> <a href="group_6.html#NR2" title="GROUP_6:NR.2">is_epimorphism</a>  ) ) ) &amp; ( for <font color="Olive" title="b1">f</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of <font color="Maroon" title="c3">F2</font> holds  <font color="Olive" title="b1">f</font> is   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c1">G</font> ) &amp; ( for <font color="Olive" title="b1">h</font> being  <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c1">G</font> holds <br/> ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F2</font> iff ( <font color="Olive" title="b1">h</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Olive" title="b1">h</font> <a href="group_6.html#NR2" title="GROUP_6:NR.2">is_epimorphism</a>  ) ) ) implies <font color="Maroon" title="c2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:3_1_2"/><i><font color="Green" title="E3">A6</font></i>: 
( ( for <font color="Olive" title="b1">f</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of <font color="Maroon" title="c2">F1</font> holds  <font color="Olive" title="b1">f</font> is   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c1">G</font> ) &amp; ( for <font color="Olive" title="b1">h</font> being  <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c1">G</font> holds <br/> ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> iff ( <font color="Olive" title="b1">h</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Olive" title="b1">h</font> <a href="group_6.html#NR2" title="GROUP_6:NR.2">is_epimorphism</a>  ) ) ) )
 <b>and </b><br/><a NAME="E2:3_1_2"/><i><font color="Green" title="E4">A7</font></i>: 
( ( for <font color="Olive" title="b1">f</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of <font color="Maroon" title="c3">F2</font> holds  <font color="Olive" title="b1">f</font> is   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c1">G</font> ) &amp; ( for <font color="Olive" title="b1">h</font> being  <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c1">G</font> holds <br/> ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F2</font> iff ( <font color="Olive" title="b1">h</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Olive" title="b1">h</font> <a href="group_6.html#NR2" title="GROUP_6:NR.2">is_epimorphism</a>  ) ) ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F2</font></span><br/>

<a NAME="E3:3_1_2"/><i><font color="Green" title="E5">A8</font></i>: 
<font color="Maroon" title="c2">F1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F2</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" title="c4">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> or <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F2</font> )</span><br/>

<b>assume </b><a NAME="E1:3_1_2_1"/><i><font color="Green" title="E6">A9</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F2</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c5">h1</font> = <font color="Maroon" title="c4">q</font> as   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="autgroup.html#E1:3_1_2"><i><font color="Green" title="E3">A6</font></i></a></i>;<br/>
<a NAME="E3:3_1_2_1"/>
( <font color="Maroon" title="c5">h1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c5">h1</font> <a href="group_6.html#NR2" title="GROUP_6:NR.2">is_epimorphism</a>  )
 
<b>by </b><i><a class="txt" href="autgroup.html#E1:3_1_2"><i><font color="Green" title="E3">A6</font></i></a>, <a class="txt" href="autgroup.html#E1:3_1_2_1"><i><font color="Green" title="E6">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:3_1_2_1"/>
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F2</font>
 <b>by </b><i><a class="txt" href="autgroup.html#E2:3_1_2"><i><font color="Green" title="E4">A7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:3_1_2"/>
<font color="Maroon" title="c3">F2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F1</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" title="c4">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F2</font> or <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font> )</span><br/>

<b>assume </b><a NAME="E1:3_1_2_2"/><i><font color="Green" title="E6">A10</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c5">h1</font> = <font color="Maroon" title="c4">q</font> as   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="autgroup.html#E2:3_1_2"><i><font color="Green" title="E4">A7</font></i></a></i>;<br/>
<a NAME="E3:3_1_2_2"/>
( <font color="Maroon" title="c5">h1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c5">h1</font> <a href="group_6.html#NR2" title="GROUP_6:NR.2">is_epimorphism</a>  )
 
<b>by </b><i><a class="txt" href="autgroup.html#E2:3_1_2"><i><font color="Green" title="E4">A7</font></i></a>, <a class="txt" href="autgroup.html#E1:3_1_2_2"><i><font color="Green" title="E6">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:3_1_2_2"/>
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F1</font>
 <b>by </b><i><a class="txt" href="autgroup.html#E1:3_1_2"><i><font color="Green" title="E3">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:3_1_2"/>
<font color="Maroon" title="c2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F2</font>
 <b>by </b><i><a class="txt" href="autgroup.html#E3:3_1_2"><i><font color="Green" title="E5">A8</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
