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<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">G</font>, <font color="Maroon" title="c3">H</font> be  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a> <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">N</font>, <font color="Maroon" title="c5">L</font>, <font color="Maroon" title="c6">G'</font> be  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font>;<br/><b>let </b><font color="Maroon" title="c7">f</font> be   <a href="group_9.html#NM4" title="GROUP_9:NM.4">Homomorphism</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font>;<br/>







<b>assume </b><a NAME="E1:154"/><i><font color="Green" title="E129">A1</font></i>: 
<font color="Maroon" title="c4">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c7">f</font>
 ;<br/>

<b>assume </b><a NAME="E2:154"/><i><font color="Green" title="E130">A2</font></i>: 
<font color="Maroon" title="c5">L</font> is  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a> <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c6">G'</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c8">Ns</font> =  <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 <font color="Maroon" title="c4">N</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c4">N</font> #) as  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a> <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="txt" href="group_9.html#E1:154"><i><font color="Green" title="E129">A1</font></i></a>, <a class="txt" href="group_9.html#E20"><i><font color="Green" title="E19">Lm7</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">Gs'</font> =  <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 <font color="Maroon" title="c6">G'</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c6">G'</font> #) as  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="txt" href="group_9.html#E40"><i><font color="Green" title="E38">Lm16</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">H'</font> being  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c3">H</font><b> such that </b><br/><a NAME="E6:154"/><i><font color="Green" title="E131">A3</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">H'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</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="c6">G'</font> &amp; <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">H'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">G'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font></span>)</span> &amp; ( <font color="Maroon" title="c7">f</font> <a href="group_9.html#NR3" title="GROUP_9:NR.3">is_epimorphism</a>  &amp; <font color="Maroon" title="c6">G'</font> is <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a> implies <font color="Maroon" title="c10">H'</font> is <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a> ) )
 <b>by </b><i><a class="txt" href="group_9.html#E1:154"><i><font color="Green" title="E129">A1</font></i></a>, <a class="ref" href="group_9.html#T79" target="_self" title="GROUP_9:th.79">Th79</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">f'</font> = <font color="Maroon" title="c7">f</font> <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 <font color="Maroon" title="c6">G'</font> as   <a href="group_9.html#NM4" title="GROUP_9:NM.4">Homomorphism</a> of <font color="Maroon" title="c6">G'</font>,<font color="Maroon" title="c10">H'</font> <b>by </b><i><a class="txt" href="group_9.html#E6:154"><i><font color="Green" title="E131">A3</font></i></a>, <a class="ref" href="group_9.html#T89" target="_self" title="GROUP_9:th.89">Th89</a></i>;<br/>
<a NAME="E8:154"/><i><font color="Green" title="E132">A4</font></i>: 
<font color="Maroon" title="c11">f'</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="c6">G'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</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="c6">G'</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T162" title="RELAT_1:th.162">RELAT_1:162</a></i>;<br/>
<a NAME="E9:154"/>
<font color="Maroon" title="c10">H'</font> is  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c10">H'</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E11"><i><font color="Green" title="E11">Lm4</font></i></a></i>;<br/>
<a NAME="E10:154"/><b>then </b><i><font color="Green" title="E133">A5</font></i>: 
 <a href="group_9.html#K14" title="GROUP_9:func.14">Image</a> <font color="Maroon" title="c11">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H'</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E6:154"><i><font color="Green" title="E131">A3</font></i></a>, <a class="txt" href="group_9.html#E8:154"><i><font color="Green" title="E132">A4</font></i></a>, <a class="ref" href="group_9.html#D25" target="_self" title="GROUP_9:def.25">Def25</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">L'</font> = <font color="Maroon" title="c5">L</font> as  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c6">G'</font> <b>by </b><i><a class="txt" href="group_9.html#E2:154"><i><font color="Green" title="E130">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">N'</font> = <font color="Maroon" title="c6">G'</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c4">N</font> as    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c6">G'</font> <b>by </b><i><a class="txt" href="group_9.html#E104"><i><font color="Green" title="E93">Lm34</font></i></a></i>;<br/>
<div><i><font color="Green" title="E134">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="154_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:154_1"/>
<font color="Maroon" title="c14">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 <font color="Maroon" title="c13">N'</font>
 ;<br/><a NAME="E2:154_1"/><b>then </b><i><font color="Green" title="E135">A7</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c6">G'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c4">N</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_9.html#D28" target="_self" title="GROUP_9:def.28">Def28</a></i>;<br/><a NAME="E3:154_1"/><b>then </b>
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c6">G'</font> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c4">N</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:154_1"/><b>then </b><i><font color="Green" title="E136">A8</font></i>: 
( <font color="Maroon" title="c14">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 <font color="Maroon" title="c6">G'</font> &amp; <font color="Maroon" title="c14">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">N</font> )
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">a'</font> = <font color="Maroon" title="c14">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">G'</font> <b>by </b><i><a class="txt" href="group_9.html#E2:154_1"><i><font color="Green" title="E135">A7</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">a''</font> = <font color="Maroon" title="c15">a'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="group_9.html#T2" target="_self" title="GROUP_9:th.2">Th2</a></i>;<br/><a NAME="E7:154_1"/>
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">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="c3">H</font>
 <b>by </b><i><a class="txt" href="group_9.html#E1:154"><i><font color="Green" title="E129">A1</font></i></a>, <a class="txt" href="group_9.html#E4:154_1"><i><font color="Green" title="E136">A8</font></i></a>, <a class="ref" href="group_9.html#T47" target="_self" title="GROUP_9:th.47">Th47</a></i>;<br/><a NAME="E8:154_1"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">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="c10">H'</font>
 <b>by </b><i><a class="ref" href="group_9.html#T4" target="_self" title="GROUP_9:th.4">Th4</a></i>;<br/><a NAME="E9:154_1"/><b>then </b>
<font color="Maroon" title="c11">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">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="c10">H'</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E10:154_1"/>
<font color="Maroon" title="c14">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="c6">G'</font> : <font color="Maroon" title="c11">f'</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="c10">H'</font> </span>}</span> 
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="154_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:154_2"/>
<font color="Maroon" title="c14">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="c6">G'</font> : <font color="Maroon" title="c11">f'</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="c10">H'</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c15">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">G'</font><b> such that </b><br/><a NAME="E3:154_2"/><i><font color="Green" title="E135">A9</font></i>: 
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a</font> &amp; <font color="Maroon" title="c11">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">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="c10">H'</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c16">a</font> = <font color="Maroon" title="c15">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="group_9.html#T2" target="_self" title="GROUP_9:th.2">Th2</a></i>;<br/><a NAME="E5:154_2"/>
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">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="c10">H'</font>
 <b>by </b><i><a class="txt" href="group_9.html#E3:154_2"><i><font color="Green" title="E135">A9</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><a NAME="E6:154_2"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">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="c3">H</font>
 <b>by </b><i><a class="ref" href="group_9.html#T4" target="_self" title="GROUP_9:th.4">Th4</a></i>;<br/><a NAME="E7:154_2"/><b>then </b>
( <font color="Maroon" title="c14">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 <font color="Maroon" title="c6">G'</font> &amp; <font color="Maroon" title="c14">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">N</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E1:154"><i><font color="Green" title="E129">A1</font></i></a>, <a class="txt" href="group_9.html#E3:154_2"><i><font color="Green" title="E135">A9</font></i></a>, <a class="ref" href="group_9.html#T47" target="_self" title="GROUP_9:th.47">Th47</a></i>;<br/><a NAME="E8:154_2"/><b>then </b>
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c6">G'</font> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c4">N</font> )
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E9:154_2"/><b>then </b>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c6">G'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c4">N</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E10:154_2"/>
<font color="Maroon" title="c14">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 <font color="Maroon" title="c13">N'</font>
 <b>by </b><i><a class="ref" href="group_9.html#D28" target="_self" title="GROUP_9:def.28">Def28</a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:154"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c13">N'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c6">G'</font> : <font color="Maroon" title="c11">f'</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="c10">H'</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="group_9.html#E13:154"><i><font color="Green" title="E134">A6</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<a NAME="E16:154"/><b>then </b><i><font color="Green" title="E135">A10</font></i>: 
<font color="Maroon" title="c13">N'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c11">f'</font>
 
<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="c14">H''</font> being  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c10">H'</font><b> such that </b><br/><a NAME="E18:154"/><i><font color="Green" title="E136">A11</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c14">H''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f'</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="c12">L'</font> &amp; <font color="Maroon" title="c11">f'</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c14">H''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c12">L'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c13">N'</font></span>)</span> &amp; ( <font color="Maroon" title="c11">f'</font> <a href="group_9.html#NR3" title="GROUP_9:NR.3">is_epimorphism</a>  &amp; <font color="Maroon" title="c12">L'</font> is <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a> implies <font color="Maroon" title="c14">H''</font> is <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a> ) )
 <b>by </b><i><a class="ref" href="group_9.html#T79" target="_self" title="GROUP_9:th.79">Th79</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">N2</font> being  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c6">G'</font><b> such that </b><br/><a NAME="E20:154"/><i><font color="Green" title="E137">A12</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c15">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f'</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c14">H''</font> &amp; ( <font color="Maroon" title="c14">H''</font> is <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a> implies ( <font color="Maroon" title="c13">N'</font> is  <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c15">N2</font> &amp; <font color="Maroon" title="c15">N2</font> is <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a> ) ) )
 <b>by </b><i><a class="txt" href="group_9.html#E16:154"><i><font color="Green" title="E135">A10</font></i></a>, <a class="ref" href="group_9.html#T78" target="_self" title="GROUP_9:th.78">Th78</a></i>;<br/>
<a NAME="E21:154"/><i><font color="Green" title="E138">A13</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">H'</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E10:154"><i><font color="Green" title="E133">A5</font></i></a>, <a class="ref" href="group_9.html#T49" target="_self" title="GROUP_9:th.49">Th49</a></i>;<br/>
<a NAME="E22:154"/><i><font color="Green" title="E139">A14</font></i>: 
<font color="Maroon" title="c15">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">L'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c13">N'</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E18:154"><i><font color="Green" title="E136">A11</font></i></a>, <a class="txt" href="group_9.html#E20:154"><i><font color="Green" title="E137">A12</font></i></a>, <a class="txt" href="group_9.html#E12"><i><font color="Green" title="E12">Lm5</font></i></a></i>;<br/>
<a NAME="E23:154"/><b>then </b><i><font color="Green" title="E140">A15</font></i>: 
<font color="Maroon" title="c15">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G'</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c4">N</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_9.html#T86" target="_self" title="GROUP_9:th.86">Th86</a></i>;<br/>
<b>thus </b><a NAME="E24:154"/>
<font color="Maroon" title="c5">L</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G'</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c4">N</font></span>)</span> is  <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c6">G'</font>
 <b>by </b><i><a class="txt" href="group_9.html#E2:154"><i><font color="Green" title="E130">A2</font></i></a>, <a class="txt" href="group_9.html#E18:154"><i><font color="Green" title="E136">A11</font></i></a>, <a class="txt" href="group_9.html#E20:154"><i><font color="Green" title="E137">A12</font></i></a>, <a class="txt" href="group_9.html#E21:154"><i><font color="Green" title="E138">A13</font></i></a>, <a class="txt" href="group_9.html#E22:154"><i><font color="Green" title="E139">A14</font></i></a>, <a class="ref" href="group_9.html#D20" target="_self" title="GROUP_9:def.20">Def20</a>, <a class="ref" href="group_9.html#T86" target="_self" title="GROUP_9:th.86">Th86</a></i>;<br/>

<b>reconsider </b><font color="Maroon" title="c16">f''</font> = <font color="Maroon" title="c7">f</font> <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="p1">(<span class="default"><font color="Maroon" title="c6">G'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font></span>)</span> 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="c6">G'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font></span>)</span>,<font color="Maroon" title="c10">H'</font> <b>by </b><i><a class="txt" href="group_9.html#E6:154"><i><font color="Green" title="E131">A3</font></i></a>, <a class="ref" href="group_9.html#T89" target="_self" title="GROUP_9:th.89">Th89</a></i>;<br/>
<a NAME="E26:154"/><i><font color="Green" title="E141">A16</font></i>: 
<font color="Maroon" title="c16">f''</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 <span class="p1">(<span class="default"><font color="Maroon" title="c6">G'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</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 <span class="p1">(<span class="default"><font color="Maroon" title="c6">G'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T162" title="RELAT_1:th.162">RELAT_1:162</a></i>;<br/>
<a NAME="E27:154"/>
<span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c9">Gs'</font></span>)</span> <a href="group_2.html#K10" title="GROUP_2:func.10">*</a> <font color="Maroon" title="c8">Ns</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Ns</font> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c9">Gs'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_3.html#T143" title="GROUP_3:th.143">GROUP_3:143</a></i>;<br/>
<a NAME="E28:154"/><b>then </b><i><font color="Green" title="E142">A17</font></i>: 
<font color="Maroon" title="c6">G'</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c4">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">N</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c6">G'</font>
 
;<br/>
<a NAME="E29:154"/><i><font color="Green" title="E143">A18</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><i><font color="Green" title="E144">A19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="154_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:154_3"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</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="c6">G'</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c18">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:154_3"/><i><font color="Green" title="E145">A20</font></i>: 
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c18">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 <font color="Maroon" title="c6">G'</font> &amp; <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">x</font> = <font color="Maroon" title="c18">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="txt" href="group_9.html#E3:154_3"><i><font color="Green" title="E145">A20</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c20">x'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:154_3"/><i><font color="Green" title="E146">A21</font></i>: 
<font color="Maroon" title="c20">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span>
 ;<br/><a NAME="E7:154_3"/><i><font color="Green" title="E147">A22</font></i>: 
<font color="Maroon" title="c20">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f</font>
 <b>by </b><i><a class="txt" href="group_9.html#E3:154_3"><i><font color="Green" title="E145">A20</font></i></a>, <a class="txt" href="group_9.html#E6:154_3"><i><font color="Green" title="E146">A21</font></i></a>, <a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>;<br/><a NAME="E8:154_3"/>
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">H</font>
 <b>by </b><i><a class="txt" href="group_9.html#E36"><i><font color="Green" title="E34">Lm13</font></i></a></i>;<br/><a NAME="E9:154_3"/><b>then </b>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c7">f</font>
 <b>by </b><i><a class="ref" href="group_9.html#T47" target="_self" title="GROUP_9:th.47">Th47</a></i>;<br/><a NAME="E10:154_3"/><b>then </b>
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c9">Gs'</font> &amp;  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c4">N</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E1:154"><i><font color="Green" title="E129">A1</font></i></a>, <a class="txt" href="group_9.html#E3:154_3"><i><font color="Green" title="E145">A20</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E11:154_3"/><b>then </b><i><font color="Green" title="E148">A23</font></i>: 
<font color="Maroon" title="c20">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">G'</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c4">N</font>
 <b>by </b><i><a class="txt" href="group_9.html#E6:154_3"><i><font color="Green" title="E146">A21</font></i></a></i>;<br/><font color="Maroon" title="c17">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">H</font></span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E3:154_3"><i><font color="Green" title="E145">A20</font></i></a>, <a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x</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="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E36"><i><font color="Green" title="E34">Lm13</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x'</font>
<b>by </b><i><a class="txt" href="group_9.html#E6:154_3"><i><font color="Green" title="E146">A21</font></i></a>, <a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></i>
;<br/><b>hence </b><a NAME="E13:154_3"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G'</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c4">N</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E7:154_3"><i><font color="Green" title="E147">A22</font></i></a>, <a class="txt" href="group_9.html#E11:154_3"><i><font color="Green" title="E148">A23</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="154_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:154_4"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G'</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c4">N</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c18">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:154_4"/><i><font color="Green" title="E145">A24</font></i>: 
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">G'</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">x</font> = <font color="Maroon" title="c18">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="txt" href="group_9.html#E3:154_4"><i><font color="Green" title="E145">A24</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c20">g1</font>, <font color="Maroon" title="c21">g2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E6:154_4"/><i><font color="Green" title="E146">A25</font></i>: 
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c20">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c6">G'</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c4">N</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E3:154_4"><i><font color="Green" title="E145">A24</font></i></a></i>;<br/><a NAME="E7:154_4"/><i><font color="Green" title="E147">A26</font></i>: 
( <font color="Maroon" title="c20">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="c6">G'</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">N</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E6:154_4"><i><font color="Green" title="E146">A25</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><font color="Maroon" title="c17">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">g1</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="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">g2</font></span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E3:154_4"><i><font color="Green" title="E145">A24</font></i></a>, <a class="txt" href="group_9.html#E6:154_4"><i><font color="Green" title="E146">A25</font></i></a>, <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="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">g1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">H</font></span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E1:154"><i><font color="Green" title="E129">A1</font></i></a>, <a class="txt" href="group_9.html#E7:154_4"><i><font color="Green" title="E147">A26</font></i></a>, <a class="ref" href="group_9.html#T47" target="_self" title="GROUP_9:th.47">Th47</a></i>
<br/>.= 
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">g1</font>
<b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>
;<br/><b>hence </b><a NAME="E9:154_4"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</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="c6">G'</font>
 <b>by </b><i><a class="txt" href="group_9.html#E29:154"><i><font color="Green" title="E143">A18</font></i></a>, <a class="txt" href="group_9.html#E6:154_4"><i><font color="Green" title="E146">A25</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/></div><b>end;</b></div>
<a NAME="E32:154"/><b>then </b>
<font color="Maroon" title="c7">f</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="c6">G'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G'</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c4">N</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_9.html#E30:154"><i><font color="Green" title="E144">A19</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<a NAME="E33:154"/><b>then </b><i><font color="Green" title="E145">A27</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">H'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</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 <span class="p1">(<span class="default"><font color="Maroon" title="c6">G'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_9.html#E6:154"><i><font color="Green" title="E131">A3</font></i></a>, <a class="txt" href="group_9.html#E28:154"><i><font color="Green" title="E142">A17</font></i></a>, <a class="ref" href="group_9.html#T30" target="_self" title="GROUP_9:th.30">Th30</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">H''</font> = <font color="Maroon" title="c14">H''</font> as  <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c10">H'</font> <b>by </b><i><a class="txt" href="group_9.html#E2:154"><i><font color="Green" title="E130">A2</font></i></a>, <a class="txt" href="group_9.html#E18:154"><i><font color="Green" title="E136">A11</font></i></a>, <a class="txt" href="group_9.html#E21:154"><i><font color="Green" title="E138">A13</font></i></a>, <a class="ref" href="group_9.html#D20" target="_self" title="GROUP_9:def.20">Def20</a></i>;<br/>
<b>set </b><font color="Maroon" title="c18">l</font> =  <a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <font color="Maroon" title="c17">H''</font>;<br/>
<b>set </b><font color="Maroon" title="c19">f1</font> = <span class="p1">(<span class="default"><a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <font color="Maroon" title="c17">H''</font></span>)</span> <a href="group_9.html#K11" title="GROUP_9:func.11">*</a> <font color="Maroon" title="c16">f''</font>;<br/>
<a NAME="E35:154"/><i><font color="Green" title="E146">A28</font></i>: 
<font color="Maroon" title="c5">L</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c6">G'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E2:154"><i><font color="Green" title="E130">A2</font></i></a>, <a class="ref" href="group_9.html#T38" target="_self" title="GROUP_9:th.38">Th38</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">N1</font> = <font color="Maroon" title="c5">L</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font> as    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c6">G'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font> <b>by </b><i><a class="txt" href="group_9.html#E2:154"><i><font color="Green" title="E130">A2</font></i></a>, <a class="ref" href="group_9.html#T38" target="_self" title="GROUP_9:th.38">Th38</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">L''</font> = <font color="Maroon" title="c5">L</font> as    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">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/>
<a NAME="E38:154"/>
<span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c21">L''</font></span>)</span> <a href="group_2.html#K10" title="GROUP_2:func.10">*</a> <font color="Maroon" title="c8">Ns</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Ns</font> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c21">L''</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_3.html#T143" title="GROUP_3:th.143">GROUP_3:143</a></i>;<br/>
<a NAME="E39:154"/><b>then </b><i><font color="Green" title="E147">A29</font></i>: 
<font color="Maroon" title="c5">L</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c4">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">N</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c5">L</font>
 
;<br/>
<a NAME="E40:154"/>
<font color="Maroon" title="c5">L</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c6">G'</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E2:154"><i><font color="Green" title="E130">A2</font></i></a>, <a class="ref" href="group_9.html#D7" target="_self" title="GROUP_9:def.7">Def7</a></i>;<br/>
<a NAME="E41:154"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">G'</font>
 
<b>by </b><i><a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></i>;<br/>
<a NAME="E42:154"/><b>then </b><i><font color="Green" title="E148">A30</font></i>: 
<font color="Maroon" title="c11">f'</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">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</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">L</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T162" title="RELAT_1:th.162">RELAT_1:162</a></i>;<br/>
<div><i><font color="Green" title="E149">A31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="154_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:154_5"/>
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">f''</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</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">L</font></span>)</span>
 ;<br/><a NAME="E2:154_5"/><b>then </b><i><font color="Green" title="E150">A32</font></i>: 
<font color="Maroon" title="c22">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"><font color="Maroon" title="c6">G'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f</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">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T139" title="FUNCT_1:th.139">FUNCT_1:139</a></i>;<br/><a NAME="E3:154_5"/><b>then </b>
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</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">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:154_5"/><b>then </b>
( <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</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">L</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c23">g1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:154_5"/><i><font color="Green" title="E151">A33</font></i>: 
( <font color="Maroon" title="c23">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> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c23">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">L</font> &amp; <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">g1</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c24">g1</font> = <font color="Maroon" title="c23">g1</font>, <font color="Maroon" title="c25">g2</font> = <font color="Maroon" title="c22">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="txt" href="group_9.html#E2:154_5"><i><font color="Green" title="E150">A32</font></i></a>, <a class="txt" href="group_9.html#E6:154_5"><i><font color="Green" title="E151">A33</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c26">g3</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E9:154_5"/><i><font color="Green" title="E152">A34</font></i>: 
<font color="Maroon" title="c25">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c26">g3</font>
 <b>by </b><i><a class="ref" href="group_1.html#T23" title="GROUP_1:th.23">GROUP_1:23</a></i>;<br/><a NAME="E10:154_5"/>
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">g2</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="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">g3</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E6:154_5"><i><font color="Green" title="E151">A33</font></i></a>, <a class="txt" href="group_9.html#E9:154_5"><i><font color="Green" title="E152">A34</font></i></a>, <a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></i>;<br/><a NAME="E11:154_5"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">g3</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="c3">H</font>
 <b>by </b><i><a class="ref" href="group_1.html#T15" title="GROUP_1:th.15">GROUP_1:15</a></i>;<br/><a NAME="E12:154_5"/><b>then </b>
<font color="Maroon" title="c26">g3</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c7">f</font>
 <b>by </b><i><a class="ref" href="group_9.html#T47" target="_self" title="GROUP_9:th.47">Th47</a></i>;<br/><a NAME="E13:154_5"/><b>then </b>
<font color="Maroon" title="c26">g3</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="c4">N</font>
 <b>by </b><i><a class="txt" href="group_9.html#E1:154"><i><font color="Green" title="E129">A1</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><b>hence </b><a NAME="E14:154_5"/>
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">L</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c4">N</font>
 <b>by </b><i><a class="txt" href="group_9.html#E6:154_5"><i><font color="Green" title="E151">A33</font></i></a>, <a class="txt" href="group_9.html#E9:154_5"><i><font color="Green" title="E152">A34</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="154_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:154_6"/><i><font color="Green" title="E150">A35</font></i>: 
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">L</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c4">N</font>
 ;<br/><a NAME="E2:154_6"/><b>then </b><i><font color="Green" title="E151">A36</font></i>: 
<font color="Maroon" title="c22">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"><font color="Maroon" title="c5">L</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E39:154"><i><font color="Green" title="E147">A29</font></i></a>, <a class="ref" href="group_9.html#T30" target="_self" title="GROUP_9:th.30">Th30</a></i>;<br/><a NAME="E3:154_6"/>
<font color="Maroon" title="c5">L</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c6">G'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font>
 <b>by </b><i><a class="txt" href="group_9.html#E35:154"><i><font color="Green" title="E146">A28</font></i></a>, <a class="ref" href="group_9.html#D7" target="_self" title="GROUP_9:def.7">Def7</a></i>;<br/><a NAME="E4:154_6"/><b>then </b><i><font color="Green" title="E152">A37</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">L</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">G'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></i>;<br/><a NAME="E5:154_6"/><b>then </b>
<font color="Maroon" title="c22">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">G'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font>
 <b>by </b><i><a class="txt" href="group_9.html#E2:154_6"><i><font color="Green" title="E151">A36</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E6:154_6"/><b>then </b>
<font color="Maroon" title="c22">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="group_9.html#T1" target="_self" title="GROUP_9:th.1">Th1</a></i>;<br/><a NAME="E7:154_6"/><b>then </b><i><font color="Green" title="E153">A38</font></i>: 
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f</font>
 <b>by </b><i><a class="txt" href="group_9.html#E29:154"><i><font color="Green" title="E143">A18</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c23">g1</font>, <font color="Maroon" title="c24">g2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E9:154_6"/><i><font color="Green" title="E154">A39</font></i>: 
( <font color="Maroon" title="c22">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c24">g2</font> &amp; <font color="Maroon" title="c23">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c5">L</font> &amp; <font color="Maroon" title="c24">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c4">N</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E1:154_6"><i><font color="Green" title="E150">A35</font></i></a></i>;<br/><a NAME="E10:154_6"/><i><font color="Green" title="E155">A40</font></i>: 
( <font color="Maroon" title="c23">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">L</font> &amp; <font color="Maroon" title="c24">g2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">N</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E9:154_6"><i><font color="Green" title="E154">A39</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">g1</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="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">g2</font></span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E9:154_6"><i><font color="Green" title="E154">A39</font></i></a>, <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="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">g1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">H</font></span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E1:154"><i><font color="Green" title="E129">A1</font></i></a>, <a class="txt" href="group_9.html#E10:154_6"><i><font color="Green" title="E155">A40</font></i></a>, <a class="ref" href="group_9.html#T47" target="_self" title="GROUP_9:th.47">Th47</a></i>
<br/>.= 
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">g1</font>
<b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>
;<br/><a NAME="E12:154_6"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</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">L</font>
 <b>by </b><i><a class="txt" href="group_9.html#E29:154"><i><font color="Green" title="E143">A18</font></i></a>, <a class="txt" href="group_9.html#E9:154_6"><i><font color="Green" title="E154">A39</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E13:154_6"/><b>then </b>
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</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">L</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E7:154_6"><i><font color="Green" title="E153">A38</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/><a NAME="E14:154_6"/><b>then </b>
<font color="Maroon" title="c22">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"><font color="Maroon" title="c6">G'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f</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">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E2:154_6"><i><font color="Green" title="E151">A36</font></i></a>, <a class="txt" href="group_9.html#E4:154_6"><i><font color="Green" title="E152">A37</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E15:154_6"/>
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">f''</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</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">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T139" title="FUNCT_1:th.139">FUNCT_1:139</a></i>;<br/></div><b>end;</b></div>
<a NAME="E45:154"/><b>then </b>
<font color="Maroon" title="c16">f''</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f'</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">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c4">N</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E42:154"><i><font color="Green" title="E148">A30</font></i></a>, <a class="txt" href="group_9.html#E43:154"><i><font color="Green" title="E149">A31</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<a NAME="E46:154"/><b>then </b><i><font color="Green" title="E150">A41</font></i>: 
<font color="Maroon" title="c16">f''</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c17">H''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c20">N1</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E18:154"><i><font color="Green" title="E136">A11</font></i></a>, <a class="txt" href="group_9.html#E39:154"><i><font color="Green" title="E147">A29</font></i></a>, <a class="ref" href="group_9.html#T30" target="_self" title="GROUP_9:th.30">Th30</a></i>;<br/>
<a NAME="E47:154"/><i><font color="Green" title="E151">A42</font></i>: 
<font color="Maroon" title="c16">f''</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</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> <span class="p2">(<span class="default"><a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <font color="Maroon" title="c17">H''</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">f''</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c17">H''</font>
 
<b>by </b><i><a class="ref" href="group_9.html#T48" target="_self" title="GROUP_9:th.48">Th48</a></i>;<br/>
<a NAME="E48:154"/><b>then </b><i><font color="Green" title="E152">A43</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_9.html#K13" title="GROUP_9:func.13">Ker</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <font color="Maroon" title="c17">H''</font></span>)</span> <a href="group_9.html#K11" title="GROUP_9:func.11">*</a> <font color="Maroon" title="c16">f''</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c20">N1</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E46:154"><i><font color="Green" title="E150">A41</font></i></a>, <a class="ref" href="group_9.html#T88" target="_self" title="GROUP_9:th.88">Th88</a></i>;<br/>
<a NAME="E49:154"/><i><font color="Green" title="E153">A44</font></i>: 
 <a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <font color="Maroon" title="c17">H''</font></span>)</span> <a href="group_9.html#K11" title="GROUP_9:func.11">*</a> <font color="Maroon" title="c16">f''</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">N1</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E46:154"><i><font color="Green" title="E150">A41</font></i></a>, <a class="txt" href="group_9.html#E47:154"><i><font color="Green" title="E151">A42</font></i></a>, <a class="txt" href="group_9.html#E12"><i><font color="Green" title="E12">Lm5</font></i></a>, <a class="ref" href="group_9.html#T88" target="_self" title="GROUP_9:th.88">Th88</a></i>;<br/>
<b>thus </b><a NAME="E50:154"/>
<font color="Maroon" title="c5">L</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font> is  <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c6">G'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font>
 <b>by </b><i><a class="txt" href="group_9.html#E48:154"><i><font color="Green" title="E152">A43</font></i></a>, <a class="txt" href="group_9.html#E12"><i><font color="Green" title="E12">Lm5</font></i></a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="154_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">N1'</font> be  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a> <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c6">G'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font>;<br/><b>let </b><font color="Maroon" title="c23">N2'</font> be  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a> <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c6">G'</font>;<br/><b>assume </b><a NAME="E1:154_7"/><i><font color="Green" title="E154">A45</font></i>: 
<font color="Maroon" title="c22">N1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font>
 ;<br/><b>assume </b><a NAME="E2:154_7"/><i><font color="Green" title="E155">A46</font></i>: 
<font color="Maroon" title="c23">N2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G'</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c4">N</font></span>)</span>
 ;<br/><a NAME="E3:154_7"/><i><font color="Green" title="E156">A47</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font></span>)</span> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <font color="Maroon" title="c17">H''</font></span>)</span> <a href="group_9.html#K11" title="GROUP_9:func.11">*</a> <font color="Maroon" title="c16">f''</font></span>)</span></span>)</span>, <a href="group_9.html#K14" title="GROUP_9:func.14">Image</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <font color="Maroon" title="c17">H''</font></span>)</span> <a href="group_9.html#K11" title="GROUP_9:func.11">*</a> <font color="Maroon" title="c16">f''</font></span>)</span> <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#T59" target="_self" title="GROUP_9:th.59">Th59</a></i>;<br/><b>set </b><font color="Maroon" title="c24">f2</font> = <span class="p1">(<span class="default"><a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <font color="Maroon" title="c17">H''</font></span>)</span> <a href="group_9.html#K11" title="GROUP_9:func.11">*</a> <font color="Maroon" title="c11">f'</font>;<br/><a NAME="E4:154_7"/><i><font color="Green" title="E157">A48</font></i>: 
 <a href="group_9.html#K14" title="GROUP_9:func.14">Image</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <font color="Maroon" title="c17">H''</font></span>)</span> <a href="group_9.html#K11" title="GROUP_9:func.11">*</a> <font color="Maroon" title="c11">f'</font></span>)</span>,<font color="Maroon" title="c6">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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <font color="Maroon" title="c17">H''</font></span>)</span> <a href="group_9.html#K11" title="GROUP_9:func.11">*</a> <font color="Maroon" title="c11">f'</font></span>)</span></span>)</span> <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#T59" target="_self" title="GROUP_9:th.59">Th59</a></i>;<br/><a NAME="E5:154_7"/><i><font color="Green" title="E158">A49</font></i>: 
<font color="Maroon" title="c16">f''</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 <span class="p1">(<span class="default"><font color="Maroon" title="c6">G'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f'</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="c6">G'</font>
 <b>by </b><i><a class="txt" href="group_9.html#E6:154"><i><font color="Green" title="E131">A3</font></i></a>, <a class="txt" href="group_9.html#E26:154"><i><font color="Green" title="E141">A16</font></i></a>, <a class="txt" href="group_9.html#E33:154"><i><font color="Green" title="E145">A27</font></i></a>, <a class="ref" href="relat_1.html#T162" title="RELAT_1:th.162">RELAT_1:162</a></i>;<br/><a NAME="E6:154_7"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <font color="Maroon" title="c17">H''</font></span>)</span> <a href="group_9.html#K11" title="GROUP_9:func.11">*</a> <font color="Maroon" title="c16">f''</font></span>)</span> <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 <span class="p1">(<span class="default"><font color="Maroon" title="c6">G'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <font color="Maroon" title="c17">H''</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">f''</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 <span class="p2">(<span class="default"><font color="Maroon" title="c6">G'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T159" title="RELAT_1:th.159">RELAT_1:159</a></i>;<br/><a NAME="E7:154_7"/><b>then </b><i><font color="Green" title="E159">A50</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <font color="Maroon" title="c17">H''</font></span>)</span> <a href="group_9.html#K11" title="GROUP_9:func.11">*</a> <font color="Maroon" title="c16">f''</font></span>)</span> <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 <span class="p1">(<span class="default"><font color="Maroon" title="c6">G'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</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#K12" title="GROUP_9:func.12">nat_hom</a> <font color="Maroon" title="c17">H''</font></span>)</span> <a href="group_9.html#K11" title="GROUP_9:func.11">*</a> <font color="Maroon" title="c11">f'</font></span>)</span> <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="c6">G'</font>
 <b>by </b><i><a class="txt" href="group_9.html#E5:154_7"><i><font color="Green" title="E158">A49</font></i></a>, <a class="ref" href="relat_1.html#T159" title="RELAT_1:th.159">RELAT_1:159</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <font color="Maroon" title="c17">H''</font></span>)</span> <a href="group_9.html#K11" title="GROUP_9:func.11">*</a> <font color="Maroon" title="c16">f''</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <font color="Maroon" title="c17">H''</font></span>)</span> <a href="group_9.html#K11" title="GROUP_9:func.11">*</a> <font color="Maroon" title="c16">f''</font></span>)</span> <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 <span class="p1">(<span class="default"><font color="Maroon" title="c6">G'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</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/>.= 
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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <font color="Maroon" title="c17">H''</font></span>)</span> <a href="group_9.html#K11" title="GROUP_9:func.11">*</a> <font color="Maroon" title="c11">f'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E7:154_7"><i><font color="Green" title="E159">A50</font></i></a>, <a class="ref" href="group_9.html#D25" target="_self" title="GROUP_9:def.25">Def25</a></i>
;<br/><a NAME="E9:154_7"/><b>then </b><i><font color="Green" title="E160">A51</font></i>: 
 <a href="group_9.html#K14" title="GROUP_9:func.14">Image</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <font color="Maroon" title="c17">H''</font></span>)</span> <a href="group_9.html#K11" title="GROUP_9:func.11">*</a> <font color="Maroon" title="c16">f''</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K14" title="GROUP_9:func.14">Image</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <font color="Maroon" title="c17">H''</font></span>)</span> <a href="group_9.html#K11" title="GROUP_9:func.11">*</a> <font color="Maroon" title="c11">f'</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E12"><i><font color="Green" title="E12">Lm5</font></i></a></i>;<br/><a NAME="E10:154_7"/>
<font color="Maroon" title="c11">f'</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</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> <span class="p2">(<span class="default"><a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <font color="Maroon" title="c17">H''</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f'</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c17">H''</font>
 <b>by </b><i><a class="ref" href="group_9.html#T48" target="_self" title="GROUP_9:th.48">Th48</a></i>;<br/><a NAME="E11:154_7"/><b>then </b>
<font color="Maroon" title="c23">N2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <font color="Maroon" title="c17">H''</font></span>)</span> <a href="group_9.html#K11" title="GROUP_9:func.11">*</a> <font color="Maroon" title="c11">f'</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E20:154"><i><font color="Green" title="E137">A12</font></i></a>, <a class="txt" href="group_9.html#E23:154"><i><font color="Green" title="E140">A15</font></i></a>, <a class="txt" href="group_9.html#E2:154_7"><i><font color="Green" title="E155">A46</font></i></a>, <a class="txt" href="group_9.html#E12"><i><font color="Green" title="E12">Lm5</font></i></a>, <a class="ref" href="group_9.html#T88" target="_self" title="GROUP_9:th.88">Th88</a></i>;<br/><b>hence </b><a NAME="E12:154_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font></span>)</span> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c22">N1'</font>,<font color="Maroon" title="c6">G'</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c23">N2'</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="group_9.html#E49:154"><i><font color="Green" title="E153">A44</font></i></a>, <a class="txt" href="group_9.html#E1:154_7"><i><font color="Green" title="E154">A45</font></i></a>, <a class="txt" href="group_9.html#E3:154_7"><i><font color="Green" title="E156">A47</font></i></a>, <a class="txt" href="group_9.html#E4:154_7"><i><font color="Green" title="E157">A48</font></i></a>, <a class="txt" href="group_9.html#E9:154_7"><i><font color="Green" title="E160">A51</font></i></a>, <a class="ref" href="group_9.html#T55" target="_self" title="GROUP_9:th.55">Th55</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E52:154"/>
for <font color="Olive" title="b1">N1</font> being <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a> <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c6">G'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font><br/> for <font color="Olive" title="b2">N2</font> being <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a> <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c6">G'</font>  st <font color="Olive" title="b1">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Olive" title="b2">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G'</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c4">N</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">G'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">N</font></span>)</span> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b1">N1</font>,<font color="Maroon" title="c6">G'</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Olive" title="b2">N2</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> 
 ;<br/>


</div>
