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

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">g</font> being  <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> holds <br/> ( <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="Olive" title="b1">g</font></span>)</span>, <a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Olive" title="b1">g</font> <a href="group_6.html#R2" title="GROUP_6:pred.2">are_isomorphic</a>  &amp;  ex <font color="Olive" title="b2">h</font> being  <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="Olive" title="b1">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="Olive" title="b1">g</font></span>)</span> st <br/>( <font color="Olive" title="b2">h</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; <font color="Olive" title="b1">g</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b2">h</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="Olive" title="b1">g</font></span>)</span></span>)</span> ) )</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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>  &amp;  ex <font color="Olive" title="b1">h</font> being  <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> st <br/>( <font color="Olive" title="b1">h</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; <font color="Maroon" title="c3">g</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">h</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> ) )</span><br/>





<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">defpred </span><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> ] <span class="kw">means </span>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#K13" title="GROUP_2:func.13">*</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="E1:108"/><span class="lab"><font color="Green" title="E61">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E2:108"/><span class="lab"><font color="Green" title="E62">A2</font></span>: 
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="108_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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="Maroon" title="c6">S</font>,<font color="Olive" title="b1">T</font>]</span><br/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E2:108_1"/><span class="lab"><font color="Green" title="E63">A3</font></span>: 
<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#K13" title="GROUP_2:func.13">*</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>
 <span class="kw">and </span><br/><a NAME="E3:108_1"/>
<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#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c7">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T15" target="_self" title="GROUP_6:th.15">Th15</a></span>;<br/>
<a NAME="E4:108_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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T54" target="_self" title="GROUP_6:th.54">Th54</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">T</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">S</font>,<font color="Maroon" title="c8">T</font>]</span><br/>

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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#K13" title="GROUP_2:func.13">*</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> implies <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> )</span><br/>

<span class="kw">assume </span><a NAME="E6:108_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#K13" title="GROUP_2:func.13">*</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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<a NAME="E7:108_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:108_1"><span class="lab"><font color="Green" title="E63">A3</font></span></a>, <a class="ref" href="group_2.html#T137" title="GROUP_2:th.137">GROUP_2:137</a></span>;<br/>
<span class="kw">then </span> <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T50" target="_self" title="GROUP_6:th.50">Th50</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D7" target="_self" title="GROUP_6:def.7">Def7</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T41" target="_self" title="GROUP_6:th.41">Th41</a></span>
;<br/>
<a NAME="E9:108_1"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T20" title="GROUP_1:th.20">GROUP_1:20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:108_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:108"/><span class="lab"><font color="Green" title="E63">A4</font></span>: 
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>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E2:108"><span class="lab"><font color="Green" title="E62">A2</font></span></a>);<br/></span>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E2:108_2"/><span class="lab"><font color="Green" title="E64">A5</font></span>: 
<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#K13" title="GROUP_2:func.13">*</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>
 <span class="kw">and </span><br/><a NAME="E3:108_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#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T15" target="_self" title="GROUP_6:th.15">Th15</a></span>;<br/><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E5:108_2"/><span class="lab"><font color="Green" title="E65">A6</font></span>: 
<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#K13" title="GROUP_2:func.13">*</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>
 <span class="kw">and </span><br/><a NAME="E6:108_2"/><span class="lab"><font color="Green" title="E66">A7</font></span>: 
<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#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T15" target="_self" title="GROUP_6:th.15">Th15</a></span>;<br/><a NAME="E7:108_2"/><span class="lab"><font color="Green" title="E67">A8</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:108"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="txt" href="#E5:108_2"><span class="lab"><font color="Green" title="E65">A6</font></span></a></span>;<br/><a NAME="E8:108_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:108"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="txt" href="#E2:108_2"><span class="lab"><font color="Green" title="E64">A5</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A9</font></span>: <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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:108_2"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="ref" href="group_2.html#T52" title="GROUP_2:th.52">GROUP_2:52</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D7" target="_self" title="GROUP_6:def.7">Def7</a></span>
;<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#K13" title="GROUP_2:func.13">*</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#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:108_2"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="txt" href="#E6:108_2"><span class="lab"><font color="Green" title="E66">A7</font></span></a>, <a class="ref" href="group_6.html#D4" target="_self" title="GROUP_6:def.4">Def4</a></span>
<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#K13" title="GROUP_2:func.13">*</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#K11" title="GROUP_2:func.11">*</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#K5" title="GROUP_2:func.5">*</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T12" title="GROUP_3:th.12">GROUP_3:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</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#K5" title="GROUP_2:func.5">*</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T6" target="_self" title="GROUP_6:th.6">Th6</a></span>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</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#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T128" title="GROUP_2:th.128">GROUP_2:128</a></span>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</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="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T140" title="GROUP_3:th.140">GROUP_3:140</a></span>
<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#K13" title="GROUP_2:func.13">*</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T127" title="GROUP_2:th.127">GROUP_2:127</a></span>
;<br/><span class="kw">hence </span><a NAME="E11:108_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:108"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="txt" href="#E9:108_2"><span class="lab"><font color="Green" title="E68">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D7" target="_self" title="GROUP_6:def.7">Def7</a></span>;<br/>
<a NAME="E7:108"/><span class="lab"><font color="Green" title="E64">A10</font></span>: 
<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="108_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">y1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c7">f</font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c7">f</font> or  not <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="Olive">b<sub>1</sub></font> or <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c9">y2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c7">f</font> or  not <font color="Maroon" title="c9">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c7">f</font> or  not <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> or <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:108_3"/>
( <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <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> or <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><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/>
<span class="kw">assume </span><a NAME="E3:108_3"/><span class="lab"><font color="Green" title="E65">A11</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font></span><br/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E5:108_3"/><span class="lab"><font color="Green" title="E66">A12</font></span>: 
<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#K13" title="GROUP_2:func.13">*</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>
 <span class="kw">and </span><br/><a NAME="E6:108_3"/>
<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#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c12">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T15" target="_self" title="GROUP_6:th.15">Th15</a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E8:108_3"/><span class="lab"><font color="Green" title="E67">A13</font></span>: 
<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#K13" title="GROUP_2:func.13">*</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>
 <span class="kw">and </span><br/><a NAME="E9:108_3"/>
<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#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c13">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T15" target="_self" title="GROUP_6:th.15">Th15</a></span>;<br/>
<a NAME="E10:108_3"/><span class="lab"><font color="Green" title="E68">A14</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:108"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="txt" href="#E8:108_3"><span class="lab"><font color="Green" title="E67">A13</font></span></a></span>;<br/>
<a NAME="E11:108_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="c12">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:108"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="txt" href="#E5:108_3"><span class="lab"><font color="Green" title="E66">A12</font></span></a></span>;<br/>
<a NAME="E12:108_3"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108_3"><span class="lab"><font color="Green" title="E65">A11</font></span></a>, <a class="txt" href="#E10:108_3"><span class="lab"><font color="Green" title="E68">A14</font></span></a>, <a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></span>;<br/>
<span class="kw">then </span> <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T41" target="_self" title="GROUP_6:th.41">Th41</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D7" target="_self" title="GROUP_6:def.7">Def7</a></span>
;<br/>
<a NAME="E14:108_3"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T50" target="_self" title="GROUP_6:th.50">Th50</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:108_3"/>
<font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:108_3"><span class="lab"><font color="Green" title="E66">A12</font></span></a>, <a class="txt" href="#E8:108_3"><span class="lab"><font color="Green" title="E67">A13</font></span></a>, <a class="ref" href="group_2.html#T137" title="GROUP_2:th.137">GROUP_2:137</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:108"/><span class="lab"><font color="Green" title="E65">A15</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E66">A16</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="108_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K1" title="RELSET_1:func.1">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> ) ) &amp; ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> ) )</span><br/><span class="kw">thus </span><a NAME="E1:108_4"/>
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K1" title="RELSET_1:func.1">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:108_4_1"/><span class="lab"><font color="Green" title="E67">A17</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> )</span><br/>

<span class="kw">hence </span><a NAME="E2:108_4_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108"><span class="lab"><font color="Green" title="E61">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font></span><br/>

<a NAME="E3:108_4_1"/><span class="lab"><font color="Green" title="E68">A18</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E4:108_4_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#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="txt" href="#E1:108_4_1"><span class="lab"><font color="Green" title="E67">A17</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:108_4_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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108_4_1"><span class="lab"><font color="Green" title="E68">A18</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:108_4"/><span class="lab"><font color="Green" title="E67">A19</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 <span class="kw">and </span><br/><a NAME="E3:108_4"/>
<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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font></span><br/><span class="kw">thus </span><a NAME="E4:108_4"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:108"><span class="lab"><font color="Green" title="E65">A15</font></span></a>, <a class="txt" href="#E2:108_4"><span class="lab"><font color="Green" title="E67">A19</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E10:108"/>
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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <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> or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:108_5"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">f</font></span><br/>

<a NAME="E2:108_5"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:108_5"/><span class="lab"><font color="Green" title="E67">A20</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T54" target="_self" title="GROUP_6:th.54">Th54</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">S</font> = <font color="Maroon" title="c9">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</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> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T16" target="_self" title="GROUP_6:th.16">Th16</a></span>;<br/>
<a NAME="E6:108_5"/>
( <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; 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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:108"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:108_5"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:108_5"><span class="lab"><font color="Green" title="E67">A20</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E11:108"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A21</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E12:108"/><span class="kw">then </span>
<font color="Maroon" title="c7">f</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:108"><span class="lab"><font color="Green" title="E64">A10</font></span></a>, <a class="ref" href="group_6.html#T70" target="_self" title="GROUP_6:th.70">Th70</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:108"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D15" target="_self" title="GROUP_6:def.15">Def15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">h</font> being  <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> st <br/>( <font color="Olive" title="b1">h</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; <font color="Maroon" title="c3">g</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">h</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> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c7">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">f</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; <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> )</span><br/>

<span class="kw">thus </span><a NAME="E14:108"/>
<font color="Maroon" title="c7">f</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:108"><span class="lab"><font color="Green" title="E67">A21</font></span></a>, <a class="txt" href="#E7:108"><span class="lab"><font color="Green" title="E64">A10</font></span></a>, <a class="ref" href="group_6.html#T70" target="_self" title="GROUP_6:th.70">Th70</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> implies <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> )</span><br/><span class="kw">assume </span><a NAME="E1:108_6"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/><span class="kw">then </span><span class="kw">reconsider </span><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="E3:108_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#K13" title="GROUP_2:func.13">*</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D9" target="_self" title="GROUP_6:def.9">Def9</a></span>;<br/><span class="kw">hence </span><a NAME="E4:108_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:108"><span class="lab"><font color="Green" title="E63">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E16:108"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:108"><span class="lab"><font color="Green" title="E66">A16</font></span></a>, <a class="ref" href="funct_1.html#T20" title="FUNCT_1:th.20">FUNCT_1:20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
