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

<span class="kw">let </span><font color="Maroon" title="c1">O</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">H</font>, <font color="Olive" title="b2">G</font> being   <a href="group_9.html#NM1" title="GROUP_9:NM.1">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font><br/>  for <font color="Olive" title="b3">h</font> being   <a href="group_9.html#NM3" title="GROUP_9:NM.3">Homomorphism</a> of <font color="Olive" title="b2">G</font>,<font color="Olive" title="b1">H</font> holds  <font color="Olive" title="b2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <span class="p1">(<span class="default"><a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Olive" title="b3">h</font></span>)</span>, <a href="group_9.html#K14" title="GROUP_9:func.14">Image</a> <font color="Olive" title="b3">h</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">H</font>, <font color="Maroon" title="c3">G</font> be   <a href="group_9.html#NM1" title="GROUP_9:NM.1">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">h</font> being   <a href="group_9.html#NM3" title="GROUP_9:NM.3">Homomorphism</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c2">H</font> holds  <font color="Maroon" title="c3">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <span class="p1">(<span class="default"><a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Olive" title="b1">h</font></span>)</span>, <a href="group_9.html#K14" title="GROUP_9:func.14">Image</a> <font color="Olive" title="b1">h</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">h</font> be   <a href="group_9.html#NM3" title="GROUP_9:NM.3">Homomorphism</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c2">H</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <span class="p1">(<span class="default"><a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span>, <a href="group_9.html#K14" title="GROUP_9:func.14">Image</a> <font color="Maroon" title="c4">h</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> </span><br/>







<span class="kw">reconsider </span><font color="Maroon" title="c5">G9</font> = <font color="Maroon" title="c3">G</font>, <font color="Maroon" title="c6">H9</font> = <font color="Maroon" title="c2">H</font> as   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">h9</font> = <font color="Maroon" title="c4">h</font> as   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c5">G9</font>,<font color="Maroon" title="c6">H9</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">g9</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="c5">G9</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="c7">h9</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="c7">h9</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:131"/><span class="lab"><font color="Green" title="E109">A1</font></span>: 
<font color="Maroon" title="c8">g9</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 <span class="kw">and </span><br/><a NAME="E5:131"/><span class="lab"><font color="Green" title="E110">A2</font></span>: 
<font color="Maroon" title="c7">h9</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c8">g9</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="c7">h9</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T79" title="GROUP_6:th.79">GROUP_6:79</a></span>;<br/>
<span class="lab"><font color="Green" title="E111">A3</font></span>:  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="c7">h9</font></span>)</span> = 
<font color="Maroon" title="c7">h9</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D10" title="GROUP_6:def.10">GROUP_6:def 10</a></span>
<br/>.= 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_9.html#K14" title="GROUP_9:func.14">Image</a> <font color="Maroon" title="c4">h</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D22" target="_self" title="GROUP_9:def.22">Def25</a></span>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="131_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c9">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_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span> implies <font color="Maroon" title="c9">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#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c7">h9</font></span>)</span> ) &amp; ( <font color="Maroon" title="c9">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#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c7">h9</font></span>)</span> implies <font color="Maroon" title="c9">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_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span> ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">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#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c7">h9</font></span>)</span> implies <font color="Maroon" title="c9">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_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:131_2_1"/>
<font color="Maroon" title="c9">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_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">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#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c7">h9</font></span>)</span></span><br/><a NAME="E2:131_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">G</font> : <font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <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="c2">H</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D21" target="_self" title="GROUP_9:def.21">Def24</a></span>;<br/><span class="kw">hence </span><a NAME="E3:131_2_1"/>
<font color="Maroon" title="c9">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#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c7">h9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D9" title="GROUP_6:def.9">GROUP_6:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:131_2"/>
<font color="Maroon" title="c9">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#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c7">h9</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">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_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span></span><br/><a NAME="E3:131_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">a9</font> where <font color="Olive" title="b1">a9</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">G9</font> : <font color="Maroon" title="c7">h9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a9</font> <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="c6">H9</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D9" title="GROUP_6:def.9">GROUP_6:def 9</a></span>;<br/><span class="kw">hence </span><a NAME="E4:131_2"/>
<font color="Maroon" title="c9">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_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D21" target="_self" title="GROUP_9:def.21">Def24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E8:131"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">A4</font></span>: 
 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#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c7">h9</font></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 <span class="p1">(<span class="default"><a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>;<br/>
<a NAME="E9:131"/>
 <a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c3">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D7" target="_self" title="GROUP_9:def.7">Def7</a></span>;<br/>
<a NAME="E10:131"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A5</font></span>: 
 <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</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#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c7">h9</font></span>)</span>, the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c7">h9</font></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</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_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span>, the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:131"><span class="lab"><font color="Green" title="E112">A4</font></span></a>, <a class="ref" href="group_2.html#T59" title="GROUP_2:th.59">GROUP_2:59</a></span>;<br/>
<a NAME="E11:131"/><span class="kw">then </span>
 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="c5">G9</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="c7">h9</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 <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <span class="p2">(<span class="default"><a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D14" target="_self" title="GROUP_9:def.14">Def14</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c8">g9</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <span class="p2">(<span class="default"><a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="group_9.html#K14" title="GROUP_9:func.14">Image</a> <font color="Maroon" title="c4">h</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:131"><span class="lab"><font color="Green" title="E111">A3</font></span></a></span>;<br/>
<a NAME="E13:131"/>
 <a href="group_9.html#K14" title="GROUP_9:func.14">Image</a> <font color="Maroon" title="c4">h</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D7" target="_self" title="GROUP_9:def.7">Def7</a></span>;<br/>
<a NAME="E14:131"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A6</font></span>: 
 <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</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="c7">h9</font></span>)</span>, the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</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="c7">h9</font></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</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_9.html#K14" title="GROUP_9:func.14">Image</a> <font color="Maroon" title="c4">h</font></span>)</span>, the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><a href="group_9.html#K14" title="GROUP_9:func.14">Image</a> <font color="Maroon" title="c4">h</font></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:131"><span class="lab"><font color="Green" title="E111">A3</font></span></a>, <a class="ref" href="group_2.html#T59" title="GROUP_2:th.59">GROUP_2:59</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E115">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="131_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c11">b</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="c3">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <span class="p2">(<span class="default"><a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">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="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">b</font></span>)</span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c12">b9</font> = <font color="Maroon" title="c11">b</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="c5">G9</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="c7">h9</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:131"><span class="lab"><font color="Green" title="E113">A5</font></span></a>, <a class="ref" href="group_9.html#D14" target="_self" title="GROUP_9:def.14">Def14</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">a9</font> = <font color="Maroon" title="c10">a</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="c5">G9</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="c7">h9</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:131"><span class="lab"><font color="Green" title="E113">A5</font></span></a>, <a class="ref" href="group_9.html#D14" target="_self" title="GROUP_9:def.14">Def14</a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">b</font></span>)</span> = 
<font color="Maroon" title="c8">g9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">a9</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c12">b9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:131"><span class="lab"><font color="Green" title="E113">A5</font></span></a>, <a class="ref" href="group_9.html#D15" target="_self" title="GROUP_9:def.15">Def15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">g9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">a9</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="c8">g9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">b9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D6" title="GROUP_6:def.6">GROUP_6:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">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="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:131"><span class="lab"><font color="Green" title="E114">A6</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="131_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">o</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</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="c3">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <span class="p2">(<span class="default"><a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span> holds  <font color="Maroon" title="c9">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"><span class="p3">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <span class="p4">(<span class="default"><a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_9.html#K14" title="GROUP_9:func.14">Image</a> <font color="Maroon" title="c4">h</font></span>)</span> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c11">a</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="c3">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <span class="p2">(<span class="default"><a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">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"><span class="p3">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <span class="p4">(<span class="default"><a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_9.html#K14" title="GROUP_9:func.14">Image</a> <font color="Maroon" title="c4">h</font></span>)</span> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="131_4_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:131_4_1"/>
( <font color="Maroon" title="c1">O</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c1">O</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="131_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:131_4_1_1"/><span class="lab"><font color="Green" title="E116">A8</font></span>: 
<font color="Maroon" title="c1">O</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">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"><span class="p3">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <span class="p4">(<span class="default"><a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_9.html#K14" title="GROUP_9:func.14">Image</a> <font color="Maroon" title="c4">h</font></span>)</span> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><br/><div class="add"><span class="kw">hence </span><font color="Maroon" title="c9">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"><span class="p3">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <span class="p4">(<span class="default"><a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span>)</span> = 
<font color="Maroon" title="c9">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"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p3">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <span class="p4">(<span class="default"><a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D6" target="_self" title="GROUP_9:def.6">Def6</a></span>
<br/>.= 
<font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T18" title="FUNCT_1:th.18">FUNCT_1:18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="group_9.html#K14" title="GROUP_9:func.14">Image</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T18" title="FUNCT_1:th.18">FUNCT_1:18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_9.html#K14" title="GROUP_9:func.14">Image</a> <font color="Maroon" title="c4">h</font></span>)</span> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:131_4_1_1"><span class="lab"><font color="Green" title="E116">A8</font></span></a>, <a class="ref" href="group_9.html#D6" target="_self" title="GROUP_9:def.6">Def6</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="131_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:131_4_1_2"/><span class="lab"><font color="Green" title="E116">A9</font></span>: 
 not <font color="Maroon" title="c1">O</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">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"><span class="p3">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <span class="p4">(<span class="default"><a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_9.html#K14" title="GROUP_9:func.14">Image</a> <font color="Maroon" title="c4">h</font></span>)</span> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c12">G99</font> = <font color="Maroon" title="c3">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <span class="p1">(<span class="default"><a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span> as   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> ;<br/><span class="kw">set </span><font color="Maroon" title="c13">f</font> =  the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <span class="p2">(<span class="default"><a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font>;<br/><a NAME="E3:131_4_1_2"/><span class="lab"><font color="Green" title="E117">A10</font></span>: 
 the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <span class="p2">(<span class="default"><a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <span class="p1">(<span class="default"><a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span> :  ex <font color="Olive" title="b3">g</font>, <font color="Olive" title="b4">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">G</font> st <br/>( <font color="Olive" title="b3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b4">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">g</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:131_4_1_2"><span class="lab"><font color="Green" title="E116">A9</font></span></a>, <a class="ref" href="group_9.html#D16" target="_self" title="GROUP_9:def.16">Def16</a></span>;<br/><a NAME="E4:131_4_1_2"/>
 the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <span class="p2">(<span class="default"><a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <span class="p2">(<span class="default"><a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c10">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:131_4_1_2"><span class="lab"><font color="Green" title="E116">A9</font></span></a>, <a class="ref" href="group_9.html#D6" target="_self" title="GROUP_9:def.6">Def6</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">f</font> =  the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <span class="p2">(<span class="default"><a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font> as   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c12">G99</font>,<font color="Maroon" title="c12">G99</font> ;<br/><a NAME="E6:131_4_1_2"/>
<font color="Maroon" title="c11">a</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 <font color="Maroon" title="c12">G99</font>
 ;<br/><a NAME="E7:131_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">a</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="c14">f</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="E8:131_4_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T1" title="FUNCT_1:th.1">FUNCT_1:1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">A</font>, <font color="Maroon" title="c16">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <span class="p1">(<span class="default"><a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E10:131_4_1_2"/><span class="lab"><font color="Green" title="E118">A11</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">A</font>,<font color="Maroon" title="c16">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E11:131_4_1_2"/><span class="lab"><font color="Green" title="E119">A12</font></span>: 
 ex <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">G</font> st <br/>( <font color="Olive" title="b1">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A</font> &amp; <font color="Olive" title="b2">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">B</font> &amp; <font color="Olive" title="b2">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">g1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:131_4_1_2"><span class="lab"><font color="Green" title="E117">A10</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">A</font> = <font color="Maroon" title="c15">A</font>, <font color="Maroon" title="c18">B</font> = <font color="Maroon" title="c16">B</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c7">h9</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:131"><span class="lab"><font color="Green" title="E113">A5</font></span></a>, <a class="ref" href="group_9.html#D14" target="_self" title="GROUP_9:def.14">Def14</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c19">g1</font>, <font color="Maroon" title="c20">g2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">G9</font><span class="kw"> such that </span><br/><a NAME="E14:131_4_1_2"/><span class="lab"><font color="Green" title="E120">A13</font></span>: 
<font color="Maroon" title="c19">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">A</font>
 <span class="kw">and </span><br/><a NAME="E15:131_4_1_2"/><span class="lab"><font color="Green" title="E121">A14</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">B</font>
 <span class="kw">and </span><br/><a NAME="E16:131_4_1_2"/><span class="lab"><font color="Green" title="E122">A15</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:131_4_1_2"><span class="lab"><font color="Green" title="E119">A12</font></span></a></span>;<br/><a NAME="E17:131_4_1_2"/><span class="lab"><font color="Green" title="E123">A16</font></span>: 
<font color="Maroon" title="c17">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">g1</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="c7">h9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:131_4_1_2"><span class="lab"><font color="Green" title="E120">A13</font></span></a>, <a class="txt" href="group_9.html#E24"><span class="lab"><font color="Green" title="E23">Lm9</font></span></a></span>;<br/><a NAME="E18:131_4_1_2"/>
<font color="Maroon" title="c19">g1</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 <font color="Maroon" title="c5">G9</font>
 ;<br/><a NAME="E19:131_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A17</font></span>: 
<font color="Maroon" title="c19">g1</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="c7">h9</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="E20:131_4_1_2"/>
<font color="Maroon" title="c20">g2</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 <font color="Maroon" title="c5">G9</font>
 ;<br/><a NAME="E21:131_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">A18</font></span>: 
<font color="Maroon" title="c20">g2</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="c7">h9</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/><span class="lab"><font color="Green" title="E126">A19</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_9.html#K14" title="GROUP_9:func.14">Image</a> <font color="Maroon" title="c4">h</font></span>)</span> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="group_9.html#K14" title="GROUP_9:func.14">Image</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D7" target="_self" title="GROUP_9:def.7">Def7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">g1</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="c7">h9</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:131_4_1_2"><span class="lab"><font color="Green" title="E118">A11</font></span></a>, <a class="txt" href="#E17:131_4_1_2"><span class="lab"><font color="Green" title="E123">A16</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>
;<br/><a NAME="E23:131_4_1_2"/><span class="lab"><font color="Green" title="E127">A20</font></span>: 
<font color="Maroon" title="c18">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</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="c7">h9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:131_4_1_2"><span class="lab"><font color="Green" title="E121">A14</font></span></a>, <a class="txt" href="group_9.html#E24"><span class="lab"><font color="Green" title="E23">Lm9</font></span></a></span>;<br/><a NAME="E24:131_4_1_2"/>
<font color="Maroon" title="c7">h9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">g1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:131_4_1_2"><span class="lab"><font color="Green" title="E122">A15</font></span></a>, <a class="ref" href="group_9.html#D18" target="_self" title="GROUP_9:def.18">Def18</a></span>;<br/><a NAME="E25:131_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">g9</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"><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="c7">h9</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">g9</font> <a href="group_6.html#K7" title="GROUP_6:func.7">*</a> <span class="p3">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <span class="p4">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c7">h9</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">g1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:131"><span class="lab"><font color="Green" title="E110">A2</font></span></a>, <a class="txt" href="#E21:131_4_1_2"><span class="lab"><font color="Green" title="E125">A18</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>;<br/><a NAME="E26:131_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">g9</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"><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="c7">h9</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <span class="p4">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c7">h9</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">g1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:131_4_1_2"><span class="lab"><font color="Green" title="E124">A17</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>;<br/><a NAME="E27:131_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A21</font></span>: 
<font color="Maroon" title="c8">g9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</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="c7">h9</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <span class="p4">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <font color="Maroon" title="c7">h9</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">g1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D8" title="GROUP_6:def.8">GROUP_6:def 8</a></span>;<br/><font color="Maroon" title="c9">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"><span class="p3">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <span class="p4">(<span class="default"><a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span>)</span> = 
<font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:131_4_1_2"><span class="lab"><font color="Green" title="E116">A9</font></span></a>, <a class="ref" href="group_9.html#D6" target="_self" title="GROUP_9:def.6">Def6</a></span>
<br/>.= 
<font color="Maroon" title="c8">g9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</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="c7">h9</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:131_4_1_2"><span class="lab"><font color="Green" title="E118">A11</font></span></a>, <a class="txt" href="#E23:131_4_1_2"><span class="lab"><font color="Green" title="E127">A20</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>
;<br/><span class="kw">hence </span><a NAME="E29:131_4_1_2"/>
<font color="Maroon" title="c9">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"><span class="p3">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <span class="p4">(<span class="default"><a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_9.html#K14" title="GROUP_9:func.14">Image</a> <font color="Maroon" title="c4">h</font></span>)</span> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:131_4_1_2"><span class="lab"><font color="Green" title="E126">A19</font></span></a>, <a class="txt" href="#E27:131_4_1_2"><span class="lab"><font color="Green" title="E128">A21</font></span></a>, <a class="ref" href="group_6.html#D8" title="GROUP_6:def.8">GROUP_6:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">g</font> = <font color="Maroon" title="c9">g</font> as   <a href="group_9.html#NM3" title="GROUP_9:NM.3">Homomorphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <span class="p2">(<span class="default"><a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="group_9.html#K14" title="GROUP_9:func.14">Image</a> <font color="Maroon" title="c4">h</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:131"><span class="lab"><font color="Green" title="E115">A7</font></span></a>, <a class="ref" href="group_9.html#D18" target="_self" title="GROUP_9:def.18">Def18</a>, <a class="ref" href="group_6.html#D6" title="GROUP_6:def.6">GROUP_6:def 6</a></span>;<br/>
<a NAME="E18:131"/>
<font color="Maroon" title="c10">g</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:131"><span class="lab"><font color="Green" title="E109">A1</font></span></a>, <a class="txt" href="#E6:131"><span class="lab"><font color="Green" title="E111">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:131"/>
<font color="Maroon" title="c3">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <span class="p1">(<span class="default"><a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c4">h</font></span>)</span>, <a href="group_9.html#K14" title="GROUP_9:func.14">Image</a> <font color="Maroon" title="c4">h</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:131"><span class="lab"><font color="Green" title="E109">A1</font></span></a>, <a class="ref" href="group_9.html#D19" target="_self" title="GROUP_9:def.19">Def22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
