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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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">A</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">B</font> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive" title="b2">A</font> holds <br/><span class="p1"><a href="grnilp_1.html#K1" title="GRNILP_1:func.1">[.</a><span class="default"><font color="Olive" title="b2">A</font>,<font color="Olive" title="b1">B</font></span><a href="grnilp_1.html#K1" title="GRNILP_1:func.1">.]</a></span> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive" title="b2">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">A</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">B</font> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c3">A</font> implies <span class="p1"><a href="grnilp_1.html#K1" title="GRNILP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">B</font></span><a href="grnilp_1.html#K1" title="GRNILP_1:func.1">.]</a></span> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c3">A</font> )</span><br/>





<a NAME="E1:14"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
 <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">A</font></span>)</span> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="grnilp_1.html#E12"><span class="lab"><font color="Green" title="E8">Lm1</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:14"/>
<font color="Maroon" title="c2">B</font> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="grnilp_1.html#K1" title="GRNILP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">B</font></span><a href="grnilp_1.html#K1" title="GRNILP_1:func.1">.]</a></span> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c3">A</font></span><br/>

<a NAME="E3:14"/><span class="kw">then </span>
 <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="grnilp_1.html#T11" target="_self" title="GRNILP_1:th.11">Th11</a></span>;<br/>
<a NAME="E4:14"/><span class="kw">then </span>
<span class="p1"><a href="grnilp_1.html#K1" title="GRNILP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">B</font></span><a href="grnilp_1.html#K1" title="GRNILP_1:func.1">.]</a></span> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#T41" title="GROUP_4:th.41">GROUP_4:41</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:14"/>
<span class="p1"><a href="grnilp_1.html#K1" title="GRNILP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">B</font></span><a href="grnilp_1.html#K1" title="GRNILP_1:func.1">.]</a></span> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="group_2.html#T65" title="GROUP_2:th.65">GROUP_2:65</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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