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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> being    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font><br/>  for <font color="Olive" title="b2">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b2">a</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="group_4.html#K7" title="GROUP_4:func.7">*</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Olive" title="b1">A</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="group_4.html#K7" title="GROUP_4:func.7">*</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b1">A</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Olive" title="b2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Olive" title="b2">a</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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" 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> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">a</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_4.html#K7" title="GROUP_4:func.7">*</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c2">A</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_4.html#K7" title="GROUP_4:func.7">*</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Olive" title="b1">a</font> &amp; <font color="Maroon" title="c2">A</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Olive" title="b1">a</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c3">a</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_4.html#K7" title="GROUP_4:func.7">*</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c2">A</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_4.html#K7" title="GROUP_4:func.7">*</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c2">A</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">a</font> )</span><br/>





<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c2">A</font> = 
<font color="Maroon" title="c3">a</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_4.html#K7" title="GROUP_4:func.7">*</a> <font color="Maroon" title="c2">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#T45" onmouseover="rs('group_4/T45')" onmouseout="rh()">GROUP_4:45</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c2">A</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T76" onmouseover="rs('group_2/T76')" onmouseout="rh()">GROUP_2:76</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"> ( <font color="Maroon" title="c3">a</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_4.html#K7" title="GROUP_4:func.7">*</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c2">A</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_4.html#K7" title="GROUP_4:func.7">*</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c2">A</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">a</font> )</span><br/>

<span class="kw">hence </span><a NAME="E2:7"/>
<font color="Maroon" title="c3">a</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_4.html#K7" title="GROUP_4:func.7">*</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#T45" onmouseover="rs('group_4/T45')" onmouseout="rh()">GROUP_4:45</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"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_4.html#K7" title="GROUP_4:func.7">*</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c2">A</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">a</font> )</span><br/>

<span class="kw">thus </span><a NAME="E3:7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_4.html#K7" title="GROUP_4:func.7">*</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T76" onmouseover="rs('group_2/T76')" onmouseout="rh()">GROUP_2:76</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"> <font color="Maroon" title="c2">A</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">a</font></span><br/>

<span class="kw">hence </span><a NAME="E4:7"/>
<font color="Maroon" title="c2">A</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#T46" onmouseover="rs('group_4/T46')" onmouseout="rh()">GROUP_4:46</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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