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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">  for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</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">B</font> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive" title="b1">A</font> holds <br/> <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> (<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">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="Olive" title="b1">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</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="c3">B</font> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">A</font> implies  <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">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="c2">A</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:11"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
<font color="Maroon" title="c3">B</font> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">A</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">  <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">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="c2">A</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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 <font color="Maroon" title="c4">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="c2">A</font>,<font color="Maroon" title="c3">B</font>) or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:11"/>
<font color="Maroon" title="c4">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="c2">A</font>,<font color="Maroon" title="c3">B</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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">A</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E4:11"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
( <font color="Maroon" title="c4">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="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="group_5.html#K2" title="GROUP_5:func.2">.]</a></span> &amp; <font color="Maroon" title="c5">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c6">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">B</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_5.html#T52" onmouseover="rs('group_5/T52')" onmouseout="rh()">GROUP_5:52</a></span>;<br/>
<a NAME="E5:11"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E4:11"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="group_2.html#T40" onmouseover="rs('group_2/T40')" onmouseout="rh()">GROUP_2:40</a></span>;<br/>
<a NAME="E6:11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A4</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="group_2.html#T50" onmouseover="rs('group_2/T50')" onmouseout="rh()">GROUP_2:50</a></span>;<br/>
<a NAME="E7:11"/>
( <font color="Maroon" title="c5">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c6">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E5:11"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="group_2.html#T51" onmouseover="rs('group_2/T51')" onmouseout="rh()">GROUP_2:51</a></span>;<br/>
<a NAME="E8:11"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T50" onmouseover="rs('group_2/T50')" onmouseout="rh()">GROUP_2:50</a></span>;<br/>
<a NAME="E9:11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A5</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:11"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="group_2.html#T50" onmouseover="rs('group_2/T50')" onmouseout="rh()">GROUP_2:50</a></span>;<br/>
<a NAME="E10:11"/>
<span class="p1"><a href="group_5.html#K2" title="GROUP_5:func.2">[.</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="group_5.html#K2" title="GROUP_5:func.2">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" onmouseover="rs('group_1/D3')" onmouseout="rh()">GROUP_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:11"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E9:11"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</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/>


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