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<div class="add">

<b>let </b><font color="Maroon" title="c1">O</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">H</font>, <font color="Maroon" title="c3">G</font> be   <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>;<br/><b>let </b><font color="Maroon" title="c4">h</font> be   <a href="group_9.html#NM4" title="GROUP_9:NM.4">Homomorphism</a> of <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c2">H</font>;<br/>







<b>reconsider </b><font color="Maroon" title="c5">G'</font> = <font color="Maroon" title="c3">G</font>, <font color="Maroon" title="c6">H'</font> = <font color="Maroon" title="c2">H</font> as   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">h'</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">G'</font>,<font color="Maroon" title="c6">H'</font> ;<br/>
<b>consider </b><font color="Maroon" title="c8">g'</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">G'</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c7">h'</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="group_6.html#K12" title="GROUP_6:func.12">Image</a> <font color="Maroon" title="c7">h'</font></span>)</span><b> such that </b><br/><a NAME="E4:136"/><i><font color="Green" title="E112">A1</font></i>: 
( <font color="Maroon" title="c8">g'</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a>  &amp; <font color="Maroon" title="c7">h'</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c8">g'</font> <a href="group_6.html#K8" title="GROUP_6:func.8">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c7">h'</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="group_6.html#T91" title="GROUP_6:th.91">GROUP_6:91</a></i>;<br/>
<a NAME="E5:136"/><i><font color="Green" title="E113">A2</font></i>: 
 <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>
 
<b>by </b><i><a class="ref" href="group_9.html#D7" target="_self" title="GROUP_9:def.7">Def7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:136_1_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>
 ;<br/><a NAME="E2:136_1_1"/><b>then </b>
<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#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">H</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="group_9.html#D24" target="_self" title="GROUP_9:def.24">Def24</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">G</font><b> such that </b><br/><a NAME="E4:136_1_1"/><i><font color="Green" title="E114">A3</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font> &amp; <font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">H</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c11">a</font> = <font color="Maroon" title="c10">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">G'</font> ;<br/><a NAME="E6:136_1_1"/>
<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="c5">G'</font> : <font color="Maroon" title="c7">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#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c6">H'</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="group_9.html#E4:136_1_1"><i><font color="Green" title="E114">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E7:136_1_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#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c7">h'</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_6.html#D10" title="GROUP_6:def.10">GROUP_6:def 10</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:136_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#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c7">h'</font></span>)</span>
 ;<br/><a NAME="E3:136_1"/><b>then </b>
<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="c5">G'</font> : <font color="Maroon" title="c7">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#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c6">H'</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="group_6.html#D10" title="GROUP_6:def.10">GROUP_6:def 10</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">a'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">G</font><b> such that </b><br/><a NAME="E5:136_1"/><i><font color="Green" title="E114">A4</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a'</font> &amp; <font color="Maroon" title="c7">h'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c6">H'</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c11">a'</font> = <font color="Maroon" title="c10">a'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">G</font> ;<br/><a NAME="E7:136_1"/>
<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#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">H</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="group_9.html#E5:136_1"><i><font color="Green" title="E114">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E8:136_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>
 <b>by </b><i><a class="ref" href="group_9.html#D24" target="_self" title="GROUP_9:def.24">Def24</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:136"/><b>then </b>
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">Ker</a> <font color="Maroon" title="c7">h'</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>
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<a NAME="E8:136"/><b>then </b><i><font color="Green" title="E114">A5</font></i>: 
 <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</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">Ker</a> <font color="Maroon" title="c7">h'</font></span>)</span>,the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p1">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c7">h'</font></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</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="group_1.html#U1" title="GROUP_1:sel.1">mult</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> #)
 
<b>by </b><i><a class="txt" href="group_9.html#E5:136"><i><font color="Green" title="E113">A2</font></i></a>, <a class="ref" href="group_2.html#T68" title="GROUP_2:th.68">GROUP_2:68</a></i>;<br/>
<a NAME="E9:136"/><b>then </b><i><font color="Green" title="E115">A6</font></i>: 
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">G'</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c7">h'</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>
 
<b>by </b><i><a class="ref" href="group_9.html#D14" target="_self" title="GROUP_9:def.14">Def14</a></i>;<br/>
<a NAME="E10:136"/><i><font color="Green" title="E116">A7</font></i>: 
 <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>
 
<b>by </b><i><a class="ref" href="group_9.html#D7" target="_self" title="GROUP_9:def.7">Def7</a></i>;<br/>
<i><font color="Green" title="E117">A8</font></i>: 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#K12" title="GROUP_6:func.12">Image</a> <font color="Maroon" title="c7">h'</font></span>)</span> = 
<font color="Maroon" title="c7">h'</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G'</font>
<b>by </b><i><a class="ref" href="group_6.html#D11" title="GROUP_6:def.11">GROUP_6:def 11</a></i>
<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>
<b>by </b><i><a class="ref" href="group_9.html#D25" target="_self" title="GROUP_9:def.25">Def25</a></i>
;<br/>
<a NAME="E12:136"/><b>then </b><i><font color="Green" title="E118">A9</font></i>: 
 <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</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#K12" title="GROUP_6:func.12">Image</a> <font color="Maroon" title="c7">h'</font></span>)</span>,the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p1">(<span class="default"><a href="group_6.html#K12" title="GROUP_6:func.12">Image</a> <font color="Maroon" title="c7">h'</font></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</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="group_1.html#U1" title="GROUP_1:sel.1">mult</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> #)
 
<b>by </b><i><a class="txt" href="group_9.html#E10:136"><i><font color="Green" title="E116">A7</font></i></a>, <a class="ref" href="group_2.html#T68" title="GROUP_2:th.68">GROUP_2:68</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c8">g'</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> <b>by </b><i><a class="txt" href="group_9.html#E9:136"><i><font color="Green" title="E115">A6</font></i></a>, <a class="txt" href="group_9.html#E11:136"><i><font color="Green" title="E117">A8</font></i></a></i>;<br/>
<div><i><font color="Green" title="E119">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="136_3"><b>now </b></a><div class="add"><b>let </b><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>;<br/><b>let </b><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>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:136_3_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="136_3_1_1"><b>suppose </b></a><a NAME="E1:136_3_1_1"/><i><font color="Green" title="E120">A11</font></i>: 
<font color="Maroon" title="c1">O</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>hence </b><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>
<b>by </b><i><a class="ref" href="group_9.html#D6" target="_self" title="GROUP_9:def.6">Def6</a></i>
<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>
<b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
<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>
<b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
<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>
<b>by </b><i><a class="txt" href="group_9.html#E1:136_3_1_1"><i><font color="Green" title="E120">A11</font></i></a>, <a class="ref" href="group_9.html#D6" target="_self" title="GROUP_9:def.6">Def6</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_3_1_2"><b>suppose </b></a><a NAME="E1:136_3_1_2"/><i><font color="Green" title="E120">A12</font></i>: 
not <font color="Maroon" title="c1">O</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c12">G''</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/><b>set </b><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="E3:136_3_1_2"/><i><font color="Green" title="E121">A13</font></i>: 
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> 
 <b>by </b><i><a class="txt" href="group_9.html#E1:136_3_1_2"><i><font color="Green" title="E120">A12</font></i></a>, <a class="ref" href="group_9.html#D16" target="_self" title="GROUP_9:def.16">Def16</a></i>;<br/><a NAME="E4:136_3_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>
 <b>by </b><i><a class="txt" href="group_9.html#E1:136_3_1_2"><i><font color="Green" title="E120">A12</font></i></a>, <a class="ref" href="group_9.html#D6" target="_self" title="GROUP_9:def.6">Def6</a></i>;<br/><b>then reconsider </b><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">G''</font>,<font color="Maroon" title="c12">G''</font> ;<br/><a NAME="E6:136_3_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">G''</font>
 ;<br/><a NAME="E7:136_3_1_2"/><b>then </b>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">f</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E8:136_3_1_2"/><b>then </b>
<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>
 <b>by </b><i><a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><b>then consider </b><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><b> such that </b><br/><a NAME="E10:136_3_1_2"/><i><font color="Green" title="E122">A14</font></i>: 
( <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> &amp;  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> ) )
 <b>by </b><i><a class="txt" href="group_9.html#E3:136_3_1_2"><i><font color="Green" title="E121">A13</font></i></a></i>;<br/><b>reconsider </b><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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_6.html#K4" title="GROUP_6:func.4">Cosets</a> <span class="p1">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c7">h'</font></span>)</span> <b>by </b><i><a class="txt" href="group_9.html#E8:136"><i><font color="Green" title="E114">A5</font></i></a>, <a class="ref" href="group_9.html#D14" target="_self" title="GROUP_9:def.14">Def14</a></i>;<br/><b>consider </b><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">G'</font><b> such that </b><br/><a NAME="E13:136_3_1_2"/><i><font color="Green" title="E123">A15</font></i>: 
( <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> &amp; <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> &amp; <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> )
 <b>by </b><i><a class="txt" href="group_9.html#E10:136_3_1_2"><i><font color="Green" title="E122">A14</font></i></a></i>;<br/><a NAME="E14:136_3_1_2"/><i><font color="Green" title="E124">A16</font></i>: 
( <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#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c7">h'</font></span>)</span> &amp; <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#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c7">h'</font></span>)</span> )
 <b>by </b><i><a class="txt" href="group_9.html#E13:136_3_1_2"><i><font color="Green" title="E123">A15</font></i></a>, <a class="txt" href="group_9.html#E24"><i><font color="Green" title="E23">Lm9</font></i></a></i>;<br/><i><font color="Green" title="E125">A17</font></i>: <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>
<b>by </b><i><a class="txt" href="group_9.html#E1:136_3_1_2"><i><font color="Green" title="E120">A12</font></i></a>, <a class="ref" href="group_9.html#D6" target="_self" title="GROUP_9:def.6">Def6</a></i>
<br/>.= 
<font color="Maroon" title="c8">g'</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#K12" title="GROUP_2:func.12">*</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c7">h'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E10:136_3_1_2"><i><font color="Green" title="E122">A14</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a>, <a class="txt" href="group_9.html#E14:136_3_1_2"><i><font color="Green" title="E124">A16</font></i></a></i>
;<br/><i><font color="Green" title="E126">A18</font></i>: <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>
<b>by </b><i><a class="ref" href="group_9.html#D7" target="_self" title="GROUP_9:def.7">Def7</a></i>
<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>
<b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<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">g'</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#K12" title="GROUP_2:func.12">*</a> <span class="p3">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c7">h'</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E10:136_3_1_2"><i><font color="Green" title="E122">A14</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a>, <a class="txt" href="group_9.html#E14:136_3_1_2"><i><font color="Green" title="E124">A16</font></i></a></i>
;<br/><a NAME="E17:136_3_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">G'</font>
 ;<br/><a NAME="E18:136_3_1_2"/><b>then </b><i><font color="Green" title="E127">A19</font></i>: 
<font color="Maroon" title="c20">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c7">h'</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E19:136_3_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">G'</font>
 ;<br/><a NAME="E20:136_3_1_2"/><b>then </b><i><font color="Green" title="E128">A20</font></i>: 
<font color="Maroon" title="c19">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c7">h'</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E21:136_3_1_2"/>
<font color="Maroon" title="c7">h'</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">h'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">g1</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E13:136_3_1_2"><i><font color="Green" title="E123">A15</font></i></a>, <a class="ref" href="group_9.html#D18" target="_self" title="GROUP_9:def.18">Def18</a></i>;<br/><a NAME="E22:136_3_1_2"/><b>then </b>
<font color="Maroon" title="c8">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="group_6.html#K10" title="GROUP_6:func.10">nat_hom</a> <span class="p3">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c7">h'</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">g'</font> <a href="group_6.html#K8" title="GROUP_6:func.8">*</a> <span class="p3">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">nat_hom</a> <span class="p4">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c7">h'</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>
 <b>by </b><i><a class="txt" href="group_9.html#E4:136"><i><font color="Green" title="E112">A1</font></i></a>, <a class="txt" href="group_9.html#E18:136_3_1_2"><i><font color="Green" title="E127">A19</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E23:136_3_1_2"/><b>then </b>
<font color="Maroon" title="c8">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="group_6.html#K10" title="GROUP_6:func.10">nat_hom</a> <span class="p3">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c7">h'</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">g'</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#K10" title="GROUP_6:func.10">nat_hom</a> <span class="p4">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c7">h'</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>
 <b>by </b><i><a class="txt" href="group_9.html#E20:136_3_1_2"><i><font color="Green" title="E128">A20</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E24:136_3_1_2"/><b>then </b>
<font color="Maroon" title="c8">g'</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#K12" title="GROUP_2:func.12">*</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c7">h'</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">g'</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#K10" title="GROUP_6:func.10">nat_hom</a> <span class="p4">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c7">h'</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>
 <b>by </b><i><a class="ref" href="group_6.html#D9" title="GROUP_6:def.9">GROUP_6:def 9</a></i>;<br/><b>hence </b><a NAME="E25:136_3_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>
 <b>by </b><i><a class="txt" href="group_9.html#E15:136_3_1_2"><i><font color="Green" title="E125">A17</font></i></a>, <a class="txt" href="group_9.html#E16:136_3_1_2"><i><font color="Green" title="E126">A18</font></i></a>, <a class="ref" href="group_6.html#D9" title="GROUP_6:def.9">GROUP_6:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_4"><b>now </b></a><div class="add"><b>let </b><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>;<br/><b>reconsider </b><font color="Maroon" title="c12">a'</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">G'</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c7">h'</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="group_9.html#E8:136"><i><font color="Green" title="E114">A5</font></i></a>, <a class="ref" href="group_9.html#D14" target="_self" title="GROUP_9:def.14">Def14</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">b'</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">G'</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Ker</a> <font color="Maroon" title="c7">h'</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="group_9.html#E8:136"><i><font color="Green" title="E114">A5</font></i></a>, <a class="ref" href="group_9.html#D14" target="_self" title="GROUP_9:def.14">Def14</a></i>;<br/><b>thus </b><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="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c11">b</font></span>)</span> = 
<font color="Maroon" title="c8">g'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">a'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c13">b'</font></span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E8:136"><i><font color="Green" title="E114">A5</font></i></a>, <a class="ref" href="group_9.html#D15" target="_self" title="GROUP_9:def.15">Def15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">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="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b'</font></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></i>
<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="group_1.html#K1" title="GROUP_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">b</font></span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E12:136"><i><font color="Green" title="E118">A9</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c10">g</font> = <font color="Maroon" title="c9">g</font> as   <a href="group_9.html#NM4" title="GROUP_9:NM.4">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> <b>by </b><i><a class="txt" href="group_9.html#E14:136"><i><font color="Green" title="E119">A10</font></i></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#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></i>;<br/>
<a NAME="E17:136"/><i><font color="Green" title="E120">A21</font></i>: 
( <font color="Maroon" title="c8">g'</font> <a href="group_6.html#NR2" title="GROUP_6:NR.2">is_epimorphism</a>  &amp; <font color="Maroon" title="c8">g'</font> <a href="group_6.html#NR1" title="GROUP_6:NR.1">is_monomorphism</a>  )
 
<b>by </b><i><a class="txt" href="group_9.html#E4:136"><i><font color="Green" title="E112">A1</font></i></a>, <a class="ref" href="group_6.html#D14" title="GROUP_6:def.14">GROUP_6:def 14</a></i>;<br/>
<a NAME="E18:136"/><b>then </b><i><font color="Green" title="E121">A22</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">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 <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>
 
<b>by </b><i><a class="txt" href="group_9.html#E11:136"><i><font color="Green" title="E117">A8</font></i></a>, <a class="ref" href="group_6.html#D13" title="GROUP_6:def.13">GROUP_6:def 13</a></i>;<br/>
<a NAME="E19:136"/>
<font color="Maroon" title="c10">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="group_9.html#E17:136"><i><font color="Green" title="E120">A21</font></i></a>, <a class="ref" href="group_6.html#D12" title="GROUP_6:def.12">GROUP_6:def 12</a></i>;<br/>
<a NAME="E20:136"/><b>then </b>
( <font color="Maroon" title="c10">g</font> <a href="group_9.html#NR3" title="GROUP_9:NR.3">is_epimorphism</a>  &amp; <font color="Maroon" title="c10">g</font> <a href="group_9.html#NR2" title="GROUP_9:NR.2">is_monomorphism</a>  )
 
<b>by </b><i><a class="txt" href="group_9.html#E18:136"><i><font color="Green" title="E121">A22</font></i></a>, <a class="ref" href="group_9.html#D19" target="_self" title="GROUP_9:def.19">Def19</a>, <a class="ref" href="group_9.html#D20" target="_self" title="GROUP_9:def.20">Def20</a></i>;<br/>
<a NAME="E21:136"/><b>then </b>
<font color="Maroon" title="c10">g</font> <a href="group_9.html#NR4" title="GROUP_9:NR.4">is_isomorphism</a> 
 
<b>by </b><i><a class="ref" href="group_9.html#D21" target="_self" title="GROUP_9:def.21">Def21</a></i>;<br/>
<b>hence </b><a NAME="E22:136"/>
<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> 
 <b>by </b><i><a class="ref" href="group_9.html#D22" target="_self" title="GROUP_9:def.22">Def22</a></i>;<br/>


</div>
