<?xml version="1.0"?>
<div class="add">

<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">H1</font>, <font color="Olive" title="b2">H2</font>, <font color="Olive" title="b3">H3</font>, <font color="Olive" title="b4">H4</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> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Olive" title="b2">H2</font> &amp; <font color="Olive" title="b3">H3</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Olive" title="b4">H4</font> holds <br/><span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Olive" title="b1">H1</font>,<font color="Olive" title="b3">H3</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">H2</font>,<font color="Olive" title="b4">H4</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">H1</font>, <font color="Maroon" title="c3">H2</font>, <font color="Maroon" title="c4">H3</font>, <font color="Maroon" title="c5">H4</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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">H1</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c3">H2</font> &amp; <font color="Maroon" title="c4">H3</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c5">H4</font> implies <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">H3</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">H2</font>,<font color="Maroon" title="c5">H4</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span> )</span><br/>









<b>assume </b><a NAME="E1:70"/>
( <font color="Maroon" title="c2">H1</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c3">H2</font> &amp; <font color="Maroon" title="c4">H3</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c5">H4</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">H1</font>,<font color="Maroon" title="c4">H3</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">H2</font>,<font color="Maroon" title="c5">H4</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span></span><br/>

<a NAME="E2:70"/><b>then </b>
 <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">H3</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">H2</font>,<font color="Maroon" title="c5">H4</font>
 
<b>by </b><i><a class="ref" href="group_5.html#T62" target="_self" title="GROUP_5:th.62">Th62</a></i>;<br/>
<b>hence </b><a NAME="E3:70"/>
<span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c4">H3</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">H2</font>,<font color="Maroon" title="c5">H4</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span>
 <b>by </b><i><a class="ref" href="group_4.html#T41" title="GROUP_4:th.41">GROUP_4:41</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>
