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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">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/> <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <font color="Olive" title="b1">H1</font>,<font color="Olive" title="b3">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="Olive" title="b2">H2</font>,<font color="Olive" title="b4">H4</font></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  <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> )</span><br/>









<b>assume </b><a NAME="E1:60"/>
( <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">  <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></span><br/>

<a NAME="E2:60"/><b>then </b>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">H1</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="c3">H2</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">H3</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="c5">H4</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H1</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c3">H2</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">H3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c4">H3</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">H4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c5">H4</font> &amp;  <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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_5.html#K4" title="GROUP_5:func.4">commutators</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c4">H3</font></span>)</span> &amp;  <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_5.html#K4" title="GROUP_5:func.4">commutators</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c5">H4</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/>
<b>hence </b><a NAME="E3:60"/>
 <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#T55" target="_self" title="GROUP_5:th.55">Th55</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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