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<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="chain_1.html#K5" title="CHAIN_1:func.5">cells</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">G</font></span>)</span>) <b>-&gt; </b>   <a href="chain_1.html#M3" title="CHAIN_1:mode.3">Cycle</a> of <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font> =  <a href="chain_1.html#K15" title="CHAIN_1:func.15">del</a> $1;<br/>
<b>consider </b><font color="Maroon" title="c4">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="chain_1.html#K1" title="CHAIN_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="chain_1.html#K5" title="CHAIN_1:func.5">cells</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">G</font></span>)</span>, <a href="chain_1.html#K1" title="CHAIN_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="chain_1.html#K5" title="CHAIN_1:func.5">cells</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span><b> such that </b><br/><a NAME="E2:104_1_1"/><i><font color="Green" title="E67">A1</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="chain_1.html#K5" title="CHAIN_1:func.5">cells</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">G</font></span>)</span> holds  <font color="Maroon" title="c4">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> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">A</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i>
<a NAME="E3:104_1_1"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="chain_1.html#K16" title="CHAIN_1:func.16">Chains</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K1" title="CHAIN_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="chain_1.html#K5" title="CHAIN_1:func.5">cells</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">G</font></span>)</span> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="chain_1.html#K16" title="CHAIN_1:func.16">Chains</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K1" title="CHAIN_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="chain_1.html#K5" title="CHAIN_1:func.5">cells</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="chain_1.html#D17" target="_self" title="CHAIN_1:def.17">Def17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">f'</font> = <font color="Maroon" title="c4">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="chain_1.html#K16" title="CHAIN_1:func.16">Chains</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="chain_1.html#K16" title="CHAIN_1:func.16">Chains</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">A</font>, <font color="Maroon" title="c7">B</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="chain_1.html#K16" title="CHAIN_1:func.16">Chains</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">G</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c8">A'</font> = <font color="Maroon" title="c6">A</font>, <font color="Maroon" title="c9">B'</font> = <font color="Maroon" title="c7">B</font> as   <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="chain_1.html#D17" target="_self" title="CHAIN_1:def.17">Def17</a></i>;<br/><b>thus </b><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">B</font></span>)</span> = 
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A'</font> <a href="chain_1.html#K8" title="CHAIN_1:func.8">+</a> <font color="Maroon" title="c9">B'</font></span>)</span>
<b>by </b><i><a class="ref" href="chain_1.html#D17" target="_self" title="CHAIN_1:def.17">Def17</a></i>
<br/>.= 
 <a href="chain_1.html#K15" title="CHAIN_1:func.15">del</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">A'</font> <a href="chain_1.html#K8" title="CHAIN_1:func.8">+</a> <font color="Maroon" title="c9">B'</font></span>)</span>
<b>by </b><i><a class="txt" href="chain_1.html#E2:104_1_1"><i><font color="Green" title="E67">A1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="chain_1.html#K15" title="CHAIN_1:func.15">del</a> <font color="Maroon" title="c8">A'</font></span>)</span> <a href="chain_1.html#K14" title="CHAIN_1:func.14">+</a> <span class="p1">(<span class="default"><a href="chain_1.html#K15" title="CHAIN_1:func.15">del</a> <font color="Maroon" title="c9">B'</font></span>)</span>
<b>by </b><i><a class="ref" href="chain_1.html#T63" target="_self" title="CHAIN_1:th.63">Th63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="chain_1.html#K15" title="CHAIN_1:func.15">del</a> <font color="Maroon" title="c8">A'</font></span>)</span> <a href="chain_1.html#K8" title="CHAIN_1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">B'</font></span>)</span>
<b>by </b><i><a class="txt" href="chain_1.html#E2:104_1_1"><i><font color="Green" title="E67">A1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">A'</font></span>)</span> <a href="chain_1.html#K8" title="CHAIN_1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">B'</font></span>)</span>
<b>by </b><i><a class="txt" href="chain_1.html#E2:104_1_1"><i><font color="Green" title="E67">A1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">B</font></span>)</span>
<b>by </b><i><a class="ref" href="chain_1.html#D17" target="_self" title="CHAIN_1:def.17">Def17</a></i>
;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">f'</font> = <font color="Maroon" title="c5">f'</font> as    <a href="mod_4.html#M1" title="MOD_4:mode.1">Homomorphism</a> of  <a href="chain_1.html#K16" title="CHAIN_1:func.16">Chains</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">G</font>, <a href="chain_1.html#K16" title="CHAIN_1:func.16">Chains</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="mod_4.html#D19" title="MOD_4:def.19">MOD_4:def 19</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">f'</font>
;<br/>

<b>thus </b><a NAME="E7:104_1_1"/>
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="chain_1.html#K16" title="CHAIN_1:func.16">Chains</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">G</font></span>)</span><br/> for <font color="Olive" title="b2">A'</font> being  <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">G</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">A'</font> holds <br/><font color="Maroon" title="c6">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="chain_1.html#K15" title="CHAIN_1:func.15">del</a> <font color="Olive" title="b2">A'</font>
 <b>by </b><i><a class="txt" href="chain_1.html#E2:104_1_1"><i><font color="Green" title="E67">A1</font></i></a></i>;<br/>


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