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

<b>let </b><font color="Maroon">H</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">G</font>,<font color="Maroon">H</font>;<br/>





<b>set </b><font color="Maroon">I</font> = <font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p1">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">J</font> =  <a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">g</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font>;<br/>
<a NAME="E1:101"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">S</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span>  ex <font color="Olive">T</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">g</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">S</font>,<font color="Olive">T</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">S</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span>;<br/>

<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E2:101_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span> &amp; <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span> <a href="group_2.html#K13">*</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="group_6.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E3:101_1"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="txt" href="group_6.html#E1:101"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">T</font> = <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">g</font></span>)</span> <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">T</font>
;<br/>

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/>

<b>assume </b><a NAME="E5:101_1"/>
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span>
 ;<br/>

<a NAME="E6:101_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="group_2.html#T137">GROUP_2:137</a></i>;<br/>
<b>then </b> <a href="group_1.html#K2">1.</a> <font color="Maroon">H</font> = 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E8:101_1"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K3">"</a> 
 
<b>by </b><i><a class="ref" href="group_1.html#T20">GROUP_1:20</a></i>;<br/>
<b>hence </b><a NAME="E9:101_1"/>
<font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="group_1.html#T19">GROUP_1:19</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">g</font></span>)</span><b> such that </b><br/><a NAME="E3:101"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">S</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">S</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">S</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">S</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span>, <font color="Maroon">T</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E2:101_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span>
 <b>and </b><br/><a NAME="E3:101_2"/>
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span> <a href="group_2.html#K13">*</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="group_6.html#T1" target="_self">Th1</a></i>;<br/><b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:101_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span>
 <b>and </b><br/><a NAME="E6:101_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span> <a href="group_2.html#K13">*</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="group_6.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E7:101_2"/>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="txt" href="group_6.html#E6:101_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="group_6.html#E8:101_2"><i><font color="Green">E74</font></i></a>, </i>;<br/><b>then </b><i><font color="Green">E74</font></i>: <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">T</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T52">GROUP_2:52</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i/>
;<br/><a NAME="E9:101_2"/>
(  <a href="group_6.html#K7" target="_self">@</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> &amp;  <a href="group_6.html#K7" target="_self">@</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> )
 ;<br/><b>then </b><font color="Maroon">S</font> <a href="group_1.html#K1">*</a> <font color="Maroon">T</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span> <a href="group_2.html#K13">*</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="txt" href="group_6.html#E8:101_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="group_6.html#D2" target="_self">Def2</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <span class="p3">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="group_2.html#K10">*</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="group_3.html#T12">GROUP_3:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span> <a href="group_2.html#K13">*</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T128">GROUP_2:128</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_2.html#K12">*</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T140">GROUP_3:140</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T127">GROUP_2:127</a></i>
;<br/><b>hence </b><a NAME="E11:101_2"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="group_1.html#K1">*</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">T</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_6.html#E6:101_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="group_6.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="group_6.html#NM2">Homomorphism</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">g</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E6:101"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:101_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">g</font></span>)</span>
 ;<br/>

<a NAME="E2:101_3"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:101_3"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="group_6.html#E1:101"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">S</font> = <font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E6:101_3"/>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">S</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:101_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:101"/><b>then </b><i><font color="Green">E110</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E8:101"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:101_4"/>
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">S</font> = <font color="Maroon">y1</font>, <font color="Maroon">T</font> = <font color="Maroon">y2</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:101_4"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span>
 <b>and </b><br/><a NAME="E5:101_4"/>
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span> <a href="group_2.html#K13">*</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="group_6.html#T1" target="_self">Th1</a></i>;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E7:101_4"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span>
 <b>and </b><br/><a NAME="E8:101_4"/>
<font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span> <a href="group_2.html#K13">*</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="group_6.html#T1" target="_self">Th1</a></i>;<br/>
<b>assume </b><a NAME="E9:101_4"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y2</font>
 ;<br/>

<a NAME="E10:101_4"/>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="group_6.html#D2" target="_self">Def2</a>, <a class="ref" href="group_6.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E11:101_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="group_6.html#T12" target="_self">Th12</a>, <a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>;<br/>
<b>then </b> <a href="group_1.html#K2">1.</a> <font color="Maroon">H</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E13:101_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E14:101_4"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i><a class="ref" href="group_6.html#D2" target="_self">Def2</a>, <a class="ref" href="group_6.html#T11" target="_self">Th11</a>, <a class="ref" href="group_2.html#T137">GROUP_2:137</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:101"/><b>then </b>
<font color="Maroon">f</font> <a href="group_6.html#NR3" target="_self">is_isomorphism</a> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E10:101"/>
<font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p1">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span>, <a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">g</font> <a href="group_6.html#R2" target="_self">are_isomorphic</a> 
 <b>by </b><i/>;<br/>

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

<a NAME="E11:101"/><i><font color="Green">E117</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>thus </b><a NAME="E12:101"/>
<font color="Maroon">f</font> <a href="group_6.html#NR3" target="_self">is_isomorphism</a> 
 <b>by </b><i>, , </i>;<br/>

<div><i><font color="Green">E176</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="101_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:101_5"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> implies ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:101_5_1"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 ;<br/>

<b>hence </b><a NAME="E2:101_5_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="group_6.html#D2" target="_self">Def2</a></i>;<br/>

<a NAME="E3:101_5_1"/>
( <span class="p1">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="group_6.html#D2" target="_self">Def2</a>, <a class="ref" href="group_6.html#T12" target="_self">Th12</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:101_5_1"/>
<span class="p1">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:101_5"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 ;<br/><b>hence </b><a NAME="E3:101_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="group_6.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:101_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 ;<br/><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> ;<br/><a NAME="E3:101_6"/>
<span class="p1">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:101_6"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <span class="p3">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:101"/>
<font color="Maroon">g</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">f</font> <a href="group_6.html#K8" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" target="_self">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="group_6.html#T11" target="_self">Th11</a>, <a class="ref" href="funct_1.html#T20">FUNCT_1:20</a></i>;<br/>


</div>
