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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">N1</font>, <font color="Olive" title="b2">N2</font> being <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="c1">G</font> holds  <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Olive" title="b1">N1</font>,<font color="Olive" title="b2">N2</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Olive" title="b2">N2</font>,<font color="Olive" title="b1">N1</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span></span><br/><b>let </b><font color="Maroon" title="c2">N1</font>, <font color="Maroon" title="c3">N2</font> be  <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="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c2">N1</font>,<font color="Maroon" title="c3">N2</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c3">N2</font>,<font color="Maroon" title="c2">N1</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span></span><br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c2">N1</font>,<font color="Maroon" title="c3">N2</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span> implies <font color="Maroon" title="c4">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c3">N2</font>,<font color="Maroon" title="c2">N1</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span> )</span><br/><b>assume </b><a NAME="E1:73_1"/>
<font color="Maroon" title="c4">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c2">N1</font>,<font color="Maroon" title="c3">N2</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c3">N2</font>,<font color="Maroon" title="c2">N1</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span></span><br/><b>then consider </b><font color="Maroon" title="c5">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c6">I</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><br/><a NAME="E3:73_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">I</font>
 <b>and </b><br/><a NAME="E4:73_1"/><i><font color="Green" title="E39">A1</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <font color="Maroon" title="c2">N1</font>,<font color="Maroon" title="c3">N2</font>
 <b>and </b><br/><a NAME="E5:73_1"/><i><font color="Green" title="E40">A2</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c6">I</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_5.html#T69" target="_self" title="GROUP_5:th.69">Th69</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1</span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> ;<br/><b>consider </b><font color="Maroon" title="c7">F1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E7:73_1"/><i><font color="Green" title="E41">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font>
 <b>and </b><br/><a NAME="E8:73_1"/><i><font color="Green" title="E42">A4</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F1</font> holds <br/><font color="Maroon" title="c7">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></i><a NAME="E9:73_1"/><i><font color="Green" title="E43">A5</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E7:73_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  =  <a href="group_4.html#K2" title="GROUP_4:func.2">@</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><a href="group_4.html#K2" title="GROUP_4:func.2">@</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1</span>)</span></span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c8">I1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><br/><a NAME="E11:73_1"/><i><font color="Green" title="E44">A6</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">I1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font>
 <b>and </b><br/><a NAME="E12:73_1"/><i><font color="Green" title="E45">A7</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">I1</font> holds <br/><font color="Maroon" title="c8">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">k</font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></i><a NAME="E13:73_1"/><i><font color="Green" title="E46">A9</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E14:73_1"/><i><font color="Green" title="E47">A10</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="txt" href="#E7:73_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E15:73_1"/><i><font color="Green" title="E48">A11</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">I1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="txt" href="#E11:73_1"><i><font color="Green" title="E44">A6</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><b>set </b><font color="Maroon" title="c9">J</font> = <font color="Maroon" title="c7">F1</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c8">I1</font>;<br/><a NAME="E16:73_1"/><i><font color="Green" title="E49">A12</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c8">I1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c6">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font> )
 <b>by </b><i><a class="txt" href="#E7:73_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="group_4.html#D4" title="GROUP_4:def.4">GROUP_4:def 4</a></i>;<br/><a NAME="E17:73_1"/><b>then </b><i><font color="Green" title="E50">X</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c6">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c6">I</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c6">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c8">I1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> )</span><br/><b>assume </b><a NAME="E1:73_1_1"/><i><font color="Green" title="E51">A13</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c6">I</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c6">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c8">I1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><a NAME="E2:73_1_1"/><b>then </b><i><font color="Green" title="E52">A14</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="txt" href="#E17:73_1"><i><font color="Green" title="E50">X</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:73_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c8">I1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><a href="group_4.html#K2" title="GROUP_4:func.2">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">I1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c7">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c7">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  &amp; <font color="Maroon" title="c8">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">I1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> &amp;  <a href="group_4.html#K2" title="GROUP_4:func.2">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">I1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">I1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c8">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K2" title="GROUP_4:func.2">@</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><a href="group_4.html#K2" title="GROUP_4:func.2">@</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="group_4.html#K2" title="GROUP_4:func.2">@</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><a href="group_4.html#K2" title="GROUP_4:func.2">@</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_4.html#K2" title="GROUP_4:func.2">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E8:73_1"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="#E12:73_1"><i><font color="Green" title="E45">A7</font></i></a>, <a class="txt" href="#E13:73_1"><i><font color="Green" title="E46">A9</font></i></a>, <a class="txt" href="#E14:73_1"><i><font color="Green" title="E47">A10</font></i></a>, <a class="txt" href="#E15:73_1"><i><font color="Green" title="E48">A11</font></i></a>, <a class="txt" href="#E1:73_1_1"><i><font color="Green" title="E51">A13</font></i></a>, <a class="txt" href="#E17:73_1"><i><font color="Green" title="E50">X</font></i></a>, <a class="ref" href="group_4.html#D4" title="GROUP_4:def.4">GROUP_4:def 4</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c8">I1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><a href="group_4.html#K2" title="GROUP_4:func.2">@</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
<b>by </b><i><a class="ref" href="group_1.html#T70" title="GROUP_1:th.70">GROUP_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p3">(<span class="default"><a href="group_4.html#K2" title="GROUP_4:func.2">@</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
<b>by </b><i><a class="ref" href="group_1.html#T72" title="GROUP_1:th.72">GROUP_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><a href="group_4.html#K2" title="GROUP_4:func.2">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T19" title="GROUP_1:th.19">GROUP_1:19</a></i>
;<br/><b>hence </b><a NAME="E5:73_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c6">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c8">I1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="#E2:73_1_1"><i><font color="Green" title="E52">A14</font></i></a>, <a class="ref" href="group_4.html#D4" title="GROUP_4:def.4">GROUP_4:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E19:73_1"/><b>then </b><i><font color="Green" title="E51">A15</font></i>: 
<font color="Maroon" title="c7">F1</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c8">I1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c6">I</font>
 <b>by </b><i><a class="txt" href="#E16:73_1"><i><font color="Green" title="E49">A12</font></i></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></i>;<br/><a NAME="E20:73_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <font color="Maroon" title="c3">N2</font>,<font color="Maroon" title="c2">N1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F1</font> or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <font color="Maroon" title="c3">N2</font>,<font color="Maroon" title="c2">N1</font> )</span><br/>

<b>assume </b><a NAME="E1:73_1_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <font color="Maroon" title="c3">N2</font>,<font color="Maroon" title="c2">N1</font></span><br/>

<b>then consider </b><font color="Maroon" title="c11">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:73_1_2"/><i><font color="Green" title="E52">A16</font></i>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F1</font>
 <b>and </b><br/><a NAME="E4:73_1_2"/><i><font color="Green" title="E53">A17</font></i>: 
<font color="Maroon" title="c7">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">y</font> = <font color="Maroon" title="c11">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E3:73_1_2"><i><font color="Green" title="E52">A16</font></i></a></i>;<br/>
<a NAME="E6:73_1_2"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">F</font>
 
<b>by </b><i><a class="txt" href="#E7:73_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="#E3:73_1_2"><i><font color="Green" title="E52">A16</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E7:73_1_2"/><b>then </b>
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">b</font>, <font color="Maroon" title="c14">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E9:73_1_2"/><i><font color="Green" title="E54">A18</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_5.html#K2" title="GROUP_5:func.2">[.</a><span class="default"><font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c14">c</font></span><a href="group_5.html#K2" title="GROUP_5:func.2">.]</a></span>
 <b>and </b><br/><a NAME="E10:73_1_2"/><i><font color="Green" title="E55">A19</font></i>: 
( <font color="Maroon" title="c13">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">N1</font> &amp; <font color="Maroon" title="c14">c</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">N2</font> )
 <b>by </b><i><a class="txt" href="#E4:73_1"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="group_5.html#T58" target="_self" title="GROUP_5:th.58">Th58</a></i>;<br/>
<a NAME="E11:73_1_2"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">F</font>
 
<b>by </b><i><a class="txt" href="#E7:73_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="#E3:73_1_2"><i><font color="Green" title="E52">A16</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E12:73_1_2"/><b>then </b>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">y</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  &amp; <font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font> )
 
<b>by </b><i><a class="txt" href="#E8:73_1"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="#E13:73_1"><i><font color="Green" title="E46">A9</font></i></a>, <a class="txt" href="#E4:73_1_2"><i><font color="Green" title="E53">A17</font></i></a>, <a class="txt" href="#E9:73_1"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E13:73_1_2"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_5.html#K2" title="GROUP_5:func.2">[.</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c13">b</font></span><a href="group_5.html#K2" title="GROUP_5:func.2">.]</a></span>
 
<b>by </b><i><a class="txt" href="#E9:73_1_2"><i><font color="Green" title="E54">A18</font></i></a>, <a class="ref" href="group_5.html#T25" target="_self" title="GROUP_5:th.25">Th25</a></i>;<br/>
<b>hence </b><a NAME="E14:73_1_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <font color="Maroon" title="c3">N2</font>,<font color="Maroon" title="c2">N1</font>
 <b>by </b><i><a class="txt" href="#E10:73_1_2"><i><font color="Green" title="E55">A19</font></i></a>, <a class="ref" href="group_5.html#T58" target="_self" title="GROUP_5:th.58">Th58</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E21:73_1"/>
<font color="Maroon" title="c4">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c3">N2</font>,<font color="Maroon" title="c2">N1</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span>
 <b>by </b><i><a class="txt" href="#E5:73_1"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="#E7:73_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="#E11:73_1"><i><font color="Green" title="E44">A6</font></i></a>, <a class="txt" href="#E19:73_1"><i><font color="Green" title="E51">A15</font></i></a>, <a class="ref" href="group_5.html#T69" target="_self" title="GROUP_5:th.69">Th69</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:73"/>
<span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c2">N1</font>,<font color="Maroon" title="c3">N2</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c3">N2</font>,<font color="Maroon" title="c2">N1</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span>
 <b>by </b><i><a class="ref" href="group_2.html#T67" title="GROUP_2:th.67">GROUP_2:67</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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