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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">G</font> is  <a href="grnilp_1.html#V1" title="GRNILP_1:attr.1">nilpotent</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="grsolv_1.html#V1" title="GRSOLV_1:attr.1">solvable</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_1_1"/>
<font color="Maroon" title="c1">G</font> is  <a href="grnilp_1.html#V1" title="GRNILP_1:attr.1">nilpotent</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">G</font> is  <a href="grsolv_1.html#V1" title="GRSOLV_1:attr.1">solvable</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="grnilp_1.html#K2" title="GRNILP_1:func.2">the_normal_subgroups_of</a> <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E3:43_1_1"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">(Omega).</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K6" title="GROUP_2:func.6">(1).</a> <font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">F</font> holds <br/> for <font color="Olive" title="b2">G1</font>, <font color="Olive" title="b3">G2</font> being   <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="c1">G</font>  st <font color="Olive" title="b2">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> holds <br/>( <font color="Olive" title="b3">G2</font> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive" title="b2">G1</font> &amp; <font color="Olive" title="b2">G1</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p1">(<span class="default">(<font color="Olive" title="b2">G1</font>,<font color="Olive" title="b3">G2</font>) <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a></span>)</span> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of  <a href="group_5.html#K10" title="GROUP_5:func.10">center</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Olive" title="b3">G2</font></span>)</span> ) ) )
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">R</font> = <font color="Maroon" title="c2">F</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="grnilp_1.html#T18" target="_self" onmouseover="rs('grnilp_1/T18')" onmouseout="rh()">Th18</a></span>;<br/>
<a NAME="E5:43_1_1"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">R</font> &amp; <font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">R</font> holds <br/> for <font color="Olive" title="b2">H1</font>, <font color="Olive" title="b3">H2</font> being    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b2">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> holds <br/>( <font color="Olive" title="b3">H2</font> is   <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_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive" title="b2">H1</font> &amp; (  for <font color="Olive" title="b4">N</font> being   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive" title="b2">H1</font>  st <font color="Olive" title="b4">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">H2</font> holds <br/><font color="Olive" title="b2">H1</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Olive" title="b4">N</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  ) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">R</font> &amp; <font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">R</font> implies  for <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Olive" title="b2">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> holds <br/>( <font color="Olive" title="b2">H2</font> is   <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_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive" title="b1">H1</font> &amp; (  for <font color="Olive" title="b3">N</font> being   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive" title="b1">H1</font>  st <font color="Olive" title="b3">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">H2</font> holds <br/><font color="Olive" title="b1">H1</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Olive" title="b3">N</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_1_1_1"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
( <font color="Maroon" title="c4">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">R</font> &amp; <font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">R</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Olive" title="b2">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> holds <br/>( <font color="Olive" title="b2">H2</font> is   <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_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive" title="b1">H1</font> &amp; (  for <font color="Olive" title="b3">N</font> being   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive" title="b1">H1</font>  st <font color="Olive" title="b3">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">H2</font> holds <br/><font color="Olive" title="b1">H1</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Olive" title="b3">N</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  ) )</span><br/>

<a NAME="E2:43_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A4</font></span>: 
( <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> is   <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="c1">G</font> &amp; <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> is   <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="c1">G</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="grnilp_1.html#T16" target="_self" onmouseover="rs('grnilp_1/T16')" onmouseout="rh()">Th16</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">H1</font>, <font color="Maroon" title="c6">H2</font> be    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c6">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> implies ( <font color="Maroon" title="c6">H2</font> is   <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_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c5">H1</font> &amp; (  for <font color="Olive" title="b1">N</font> being   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c5">H1</font>  st <font color="Olive" title="b1">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">H2</font> holds <br/><font color="Maroon" title="c5">H1</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Olive" title="b1">N</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E3:43_1_1_1"/><span class="lab"><font color="Green" title="E31">A5</font></span>: 
( <font color="Maroon" title="c5">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c6">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">H2</font> is   <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_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c5">H1</font> &amp; (  for <font color="Olive" title="b1">N</font> being   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c5">H1</font>  st <font color="Olive" title="b1">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">H2</font> holds <br/><font color="Maroon" title="c5">H1</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Olive" title="b1">N</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  ) )</span><br/>

<a NAME="E4:43_1_1_1"/>
 for <font color="Olive" title="b1">N</font> being   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c5">H1</font>  st <font color="Olive" title="b1">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">H2</font> holds <br/><font color="Maroon" title="c5">H1</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Olive" title="b1">N</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">N</font> be   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c5">H1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">H2</font> implies <font color="Maroon" title="c5">H1</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c7">N</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_1_1_1_1"/><span class="lab"><font color="Green" title="E32">A6</font></span>: 
<font color="Maroon" title="c7">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">H2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">H1</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c7">N</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">N9</font> = <font color="Maroon" title="c7">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="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_1_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E2:43_1_1_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a></span>;<br/>
<a NAME="E3:43_1_1_1_1"/><span class="lab"><font color="Green" title="E33">A7</font></span>: 
<font color="Maroon" title="c5">H1</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c5">H1</font>,<font color="Maroon" title="c8">N9</font>) <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a></span>)</span> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of  <a href="group_5.html#K10" title="GROUP_5:func.10">center</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c8">N9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_1"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E1:43_1_1_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E3:43_1_1_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E1:43_1_1_1_1"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E2:43_1_1_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a></span>;<br/>
<a NAME="E4:43_1_1_1_1"/>
 <a href="group_5.html#K10" title="GROUP_5:func.10">center</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c8">N9</font></span>)</span> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_5.html#T80" onmouseover="rs('group_5/T80')" onmouseout="rh()">GROUP_5:80</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:43_1_1_1_1"/>
<font color="Maroon" title="c5">H1</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c7">N</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_1_1_1"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="ref" href="group_6.html#D1" onmouseover="rs('group_6/D1')" onmouseout="rh()">GROUP_6:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:43_1_1_1"/>
( <font color="Maroon" title="c6">H2</font> is   <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_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c5">H1</font> &amp; (  for <font color="Olive" title="b1">N</font> being   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c5">H1</font>  st <font color="Olive" title="b1">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">H2</font> holds <br/><font color="Maroon" title="c5">H1</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Olive" title="b1">N</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_1"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E1:43_1_1_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E3:43_1_1_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E2:43_1_1_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="group_6.html#T8" onmouseover="rs('group_6/T8')" onmouseout="rh()">GROUP_6:8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">take </span>
<font color="Maroon" title="c3">R</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="grsolv_1.html#D1" onmouseover="rs('grsolv_1/D1')" onmouseout="rh()">GRSOLV_1:def 1</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">R</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c3">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">(Omega).</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c3">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K6" title="GROUP_2:func.6">(1).</a> <font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">R</font> or  not <a href="nat_1.html#K1" title="NAT_1:func.1">K336</a>(<font color="Olive">b<sub>1</sub></font>,1) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">R</font> or  for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>  holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="nat_1.html#K1" title="NAT_1:func.1">K336</a>(<font color="Olive">b<sub>1</sub></font>,1) or ( <font color="Olive">b<sub>3</sub></font> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive">b<sub>2</sub></font> &amp; (  for <font color="Olive">b<sub>4</sub></font> being    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive">b<sub>2</sub></font>  holds <br/>(  not <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>3</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Olive">b<sub>4</sub></font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  ) ) ) ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E6:43_1_1"/>
(  not  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">R</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c3">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">(Omega).</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c3">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K6" title="GROUP_2:func.6">(1).</a> <font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">R</font> or  not <a href="nat_1.html#K1" title="NAT_1:func.1">K336</a>(<font color="Olive">b<sub>1</sub></font>,1) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">R</font> or  for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>  holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="nat_1.html#K1" title="NAT_1:func.1">K336</a>(<font color="Olive">b<sub>1</sub></font>,1) or ( <font color="Olive">b<sub>3</sub></font> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive">b<sub>2</sub></font> &amp; (  for <font color="Olive">b<sub>4</sub></font> being    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive">b<sub>2</sub></font>  holds <br/>(  not <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>3</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Olive">b<sub>4</sub></font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  ) ) ) ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_1"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E5:43_1_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
