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

<b>let </b><font color="Maroon" title="c1">H</font>, <font color="Maroon" title="c2">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c3">g</font> be   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c1">H</font>;<br/>





<b>set </b><font color="Maroon" title="c4">I</font> = <font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c5">J</font> =  <a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Maroon" title="c3">g</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span> holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font>;<br/>
<a NAME="E1:102"/><i><font color="Green" title="E65">A1</font></i>: 
for <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span>  ex <font color="Olive" title="b2">T</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Maroon" title="c3">g</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">T</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">S</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span>;<br/>

<b>consider </b><font color="Maroon" title="c7">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E2:102_1"/><i><font color="Green" title="E66">A2</font></i>: 
( <font color="Maroon" title="c6">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span> &amp; <font color="Maroon" title="c6">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c7">a</font> )
 <b>by </b><i><a class="ref" href="group_6.html#T15" target="_self" title="GROUP_6:th.15">Th15</a></i>;<br/>
<a NAME="E3:102_1"/>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Maroon" title="c3">g</font>
 
<b>by </b><i><a class="ref" href="group_6.html#T54" target="_self" title="GROUP_6:th.54">Th54</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">T</font> = <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Maroon" title="c3">g</font></span>)</span> <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">T</font>
;<br/>

<b>let </b><font color="Maroon" title="c9">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font>;<br/>

<b>assume </b><a NAME="E5:102_1"/>
<font color="Maroon" title="c6">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span>
 ;<br/>

<a NAME="E6:102_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font>
 
<b>by </b><i><a class="txt" href="group_6.html#E2:102_1"><i><font color="Green" title="E66">A2</font></i></a>, <a class="ref" href="group_2.html#T137" title="GROUP_2:th.137">GROUP_2:137</a></i>;<br/>
<b>then </b> <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">H</font> = 
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">b</font></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#T50" target="_self" title="GROUP_6:th.50">Th50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#D7" target="_self" title="GROUP_6:def.7">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#T41" target="_self" title="GROUP_6:th.41">Th41</a></i>
;<br/>
<a NAME="E8:102_1"/><b>then </b>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 
<b>by </b><i><a class="ref" href="group_1.html#T20" title="GROUP_1:th.20">GROUP_1:20</a></i>;<br/>
<b>hence </b><a NAME="E9:102_1"/>
<font color="Maroon" title="c8">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font>
 <b>by </b><i><a class="ref" href="group_1.html#T19" title="GROUP_1:th.19">GROUP_1:19</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c6">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Maroon" title="c3">g</font></span>)</span><b> such that </b><br/><a NAME="E3:102"/><i><font color="Green" title="E66">A3</font></i>: 
for <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">S</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">S</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="group_6.html#E1:102"><i><font color="Green" title="E65">A1</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">S</font>, <font color="Maroon" title="c8">T</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c9">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E2:102_2"/><i><font color="Green" title="E67">A4</font></i>: 
<font color="Maroon" title="c7">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span>
 <b>and </b><br/><a NAME="E3:102_2"/>
<font color="Maroon" title="c7">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="ref" href="group_6.html#T15" target="_self" title="GROUP_6:th.15">Th15</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E5:102_2"/><i><font color="Green" title="E68">A5</font></i>: 
<font color="Maroon" title="c8">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span>
 <b>and </b><br/><a NAME="E6:102_2"/><i><font color="Green" title="E69">A6</font></i>: 
<font color="Maroon" title="c8">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="group_6.html#T15" target="_self" title="GROUP_6:th.15">Th15</a></i>;<br/><a NAME="E7:102_2"/>
( <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font> &amp; <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font> )
 <b>by </b><i><a class="txt" href="group_6.html#E3:102"><i><font color="Green" title="E66">A3</font></i></a>, <a class="txt" href="group_6.html#E2:102_2"><i><font color="Green" title="E67">A4</font></i></a>, <a class="txt" href="group_6.html#E5:102_2"><i><font color="Green" title="E68">A5</font></i></a></i>;<br/><b>then </b><i><font color="Green" title="E70">A7</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">S</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">T</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T52" title="GROUP_2:th.52">GROUP_2:52</a></i>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#D7" target="_self" title="GROUP_6:def.7">Def7</a></i>
;<br/><font color="Maroon" title="c7">S</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">T</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="txt" href="group_6.html#E2:102_2"><i><font color="Green" title="E67">A4</font></i></a>, <a class="txt" href="group_6.html#E6:102_2"><i><font color="Green" title="E69">A6</font></i></a>, <a class="ref" href="group_6.html#D4" target="_self" title="GROUP_6:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p3">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="group_2.html#K10" title="GROUP_2:func.10">*</a> <span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="ref" href="group_3.html#T12" title="GROUP_3:th.12">GROUP_3:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="ref" href="group_6.html#T6" target="_self" title="GROUP_6:th.6">Th6</a></i>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T128" title="GROUP_2:th.128">GROUP_2:128</a></i>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T140" title="GROUP_3:th.140">GROUP_3:140</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T127" title="GROUP_2:th.127">GROUP_2:127</a></i>
;<br/><b>hence </b><a NAME="E10:102_2"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">S</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">S</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">T</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_6.html#E3:102"><i><font color="Green" title="E66">A3</font></i></a>, <a class="txt" href="group_6.html#E8:102_2"><i><font color="Green" title="E70">A7</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c6">f</font> as   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Maroon" title="c3">g</font></span>)</span> <b>by </b><i><a class="ref" href="group_6.html#D7" target="_self" title="GROUP_6:def.7">Def7</a></i>;<br/>
<a NAME="E6:102"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</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><br/>

<b>assume </b><a NAME="E1:102_3"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Maroon" title="c3">g</font></span>)</span>
 ;<br/>

<a NAME="E2:102_3"/><b>then </b>
<font color="Maroon" title="c8">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Maroon" title="c3">g</font>
 
<b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E4:102_3"/><i><font color="Green" title="E67">A8</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="ref" href="group_6.html#T54" target="_self" title="GROUP_6:th.54">Th54</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">S</font> = <font color="Maroon" title="c9">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="group_6.html#T16" target="_self" title="GROUP_6:th.16">Th16</a></i>;<br/>
<a NAME="E6:102_3"/>
( <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font> &amp; <font color="Maroon" title="c10">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f</font> )
 
<b>by </b><i><a class="txt" href="group_6.html#E3:102"><i><font color="Green" title="E66">A3</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:102_3"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">f</font>
 <b>by </b><i><a class="txt" href="group_6.html#E4:102_3"><i><font color="Green" title="E67">A8</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:102"/><b>then </b><i><font color="Green" title="E67">A9</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Maroon" title="c3">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E8:102"/><i><font color="Green" title="E68">A10</font></i>: 
<font color="Maroon" title="c7">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">y1</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="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon" title="c9">y2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:102_4"/>
( <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c9">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c10">S</font> = <font color="Maroon" title="c8">y1</font>, <font color="Maroon" title="c11">T</font> = <font color="Maroon" title="c9">y2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> ;<br/>
<b>consider </b><font color="Maroon" title="c12">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E4:102_4"/><i><font color="Green" title="E69">A11</font></i>: 
<font color="Maroon" title="c10">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span>
 <b>and </b><br/><a NAME="E5:102_4"/>
<font color="Maroon" title="c10">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c12">a</font>
 <b>by </b><i><a class="ref" href="group_6.html#T15" target="_self" title="GROUP_6:th.15">Th15</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E7:102_4"/><i><font color="Green" title="E70">A12</font></i>: 
<font color="Maroon" title="c11">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span>
 <b>and </b><br/><a NAME="E8:102_4"/>
<font color="Maroon" title="c11">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c13">b</font>
 <b>by </b><i><a class="ref" href="group_6.html#T15" target="_self" title="GROUP_6:th.15">Th15</a></i>;<br/>
<b>assume </b><a NAME="E9:102_4"/><i><font color="Green" title="E71">A13</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y2</font>
 ;<br/>

<a NAME="E10:102_4"/>
( <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font> &amp; <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b</font> )
 
<b>by </b><i><a class="txt" href="group_6.html#E3:102"><i><font color="Green" title="E66">A3</font></i></a>, <a class="txt" href="group_6.html#E4:102_4"><i><font color="Green" title="E69">A11</font></i></a>, <a class="txt" href="group_6.html#E7:102_4"><i><font color="Green" title="E70">A12</font></i></a></i>;<br/>
<a NAME="E11:102_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">H</font>
 
<b>by </b><i><a class="txt" href="group_6.html#E9:102_4"><i><font color="Green" title="E71">A13</font></i></a>, <a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></i>;<br/>
<b>then </b> <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">H</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#T41" target="_self" title="GROUP_6:th.41">Th41</a></i>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c12">a</font></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#D7" target="_self" title="GROUP_6:def.7">Def7</a></i>
;<br/>
<a NAME="E13:102_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c12">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font>
 
<b>by </b><i><a class="ref" href="group_6.html#T50" target="_self" title="GROUP_6:th.50">Th50</a></i>;<br/>
<b>hence </b><a NAME="E14:102_4"/>
<font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font>
 <b>by </b><i><a class="txt" href="group_6.html#E4:102_4"><i><font color="Green" title="E69">A11</font></i></a>, <a class="txt" href="group_6.html#E7:102_4"><i><font color="Green" title="E70">A12</font></i></a>, <a class="ref" href="group_2.html#T137" title="GROUP_2:th.137">GROUP_2:137</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:102"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a> 
 
<b>by </b><i><a class="txt" href="group_6.html#E7:102"><i><font color="Green" title="E67">A9</font></i></a>, <a class="ref" href="group_6.html#T70" target="_self" title="GROUP_6:th.70">Th70</a></i>;<br/>
<b>hence </b><a NAME="E10:102"/>
<font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span>, <a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Maroon" title="c3">g</font> <a href="group_6.html#R2" title="GROUP_6:pred.2">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="group_6.html#D15" target="_self" title="GROUP_6:def.15">Def15</a></i>;<br/>

<b>take </b>
<font color="Maroon" title="c7">f</font>
;<br/>

<a NAME="E11:102"/><i><font color="Green" title="E69">A14</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>thus </b><a NAME="E12:102"/>
<font color="Maroon" title="c7">f</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a> 
 <b>by </b><i><a class="txt" href="group_6.html#E7:102"><i><font color="Green" title="E67">A9</font></i></a>, <a class="txt" href="group_6.html#E8:102"><i><font color="Green" title="E68">A10</font></i></a>, <a class="ref" href="group_6.html#T70" target="_self" title="GROUP_6:th.70">Th70</a></i>;<br/>

<div><i><font color="Green" title="E70">A15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="102_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>thus </b><a NAME="E1:102_5"/>
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g</font> implies ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:102_5_1"/><i><font color="Green" title="E71">A16</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g</font>
 ;<br/>

<b>hence </b><a NAME="E2:102_5_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="group_6.html#E11:102"><i><font color="Green" title="E69">A14</font></i></a></i>;<br/>

<a NAME="E3:102_5_1"/>
( <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="group_6.html#E11:102"><i><font color="Green" title="E69">A14</font></i></a>, <a class="txt" href="group_6.html#E1:102_5_1"><i><font color="Green" title="E71">A16</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:102_5_1"/>
<span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f</font>
 ;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:102_5"/>
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f</font> )
 ;<br/><b>hence </b><a NAME="E3:102_5"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g</font>
 <b>by </b><i><a class="txt" href="group_6.html#E11:102"><i><font color="Green" title="E69">A14</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:102_6"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c8">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> ;<br/><a NAME="E3:102_6"/>
<span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_6.html#D9" target="_self" title="GROUP_6:def.9">Def9</a></i>;<br/><b>hence </b><a NAME="E4:102_6"/>
<font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <span class="p3">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_6.html#E3:102"><i><font color="Green" title="E66">A3</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:102"/>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="group_6.html#K7" title="GROUP_6:func.7">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="group_6.html#E13:102"><i><font color="Green" title="E70">A15</font></i></a>, <a class="ref" href="funct_1.html#T20" title="FUNCT_1:th.20">FUNCT_1:20</a></i>;<br/>


</div>
