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<div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">H</font> be    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/>



<b>thus </b><a NAME="E1:126"/>
( <font color="Maroon">H</font> is  <a href="group_3.html#V1" target="_self">normal</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> implies for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> holds  <font color="Olive">a</font> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="group_2.html#K13">*</a> <font color="Olive">a</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="126_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:126_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">H</font> is  <a href="group_3.html#V1" target="_self">normal</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>
 ;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/>

<i><font color="Green">E39</font></i>:  <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> = 
the <a href="struct_0.html#U1">carrier</a> of <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font>,the <a href="group_1.html#U1">mult</a> of <font color="Maroon">H</font> #)
<b>by </b><i><a class="ref" href="group_3.html#T53" target="_self">Th53</a>, <a class="ref" href="group_3.html#T44" target="_self">Th44</a></i>
<br/>.= 
 <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font>

;<br/>
<a NAME="E3:126_1"/><i><font color="Green">E40</font></i>: 
 <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T5" target="_self">Th5</a></i>;<br/>
<b>thus </b><font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="group_3.html#T54" target="_self">Th54</a>, <a class="ref" href="group_3.html#T56" target="_self">Th56</a>, <a class="ref" href="group_2.html#T39">GROUP_2:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="group_2.html#T127">GROUP_2:127</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>
<br/>.= 
<font color="Maroon">H</font> <a href="group_2.html#K13">*</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="group_2.html#T43">GROUP_2:43</a></i>
;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:126"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> holds  <font color="Olive">a</font> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="group_2.html#K13">*</a> <font color="Olive">a</font>
 ;<br/>

<a NAME="E3:126"/>
<font color="Maroon">H</font> is <a href="group_3.html#V1" target="_self">normal</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="126_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_3.html#D13" target="_self">GROUP_3:def 13</a><br/>

the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="group_3.html#T5" target="_self">Th5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_2.html#K13">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="group_3.html#T53" target="_self">Th53</a></i>
<br/>.= 
<font color="Maroon">H</font> <a href="group_2.html#K13">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T129">GROUP_2:129</a></i>
<br/>.= 
<font color="Maroon">H</font> <a href="group_2.html#K13">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font>
<b>by </b><i><a class="ref" href="group_2.html#T132">GROUP_2:132</a></i>
;<br/>
<b>hence </b><a NAME="E2:126_2"/>
<font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font>,the <a href="group_1.html#U1">mult</a> of <font color="Maroon">H</font> #)
 <b>by </b><i><a class="ref" href="group_2.html#T68">GROUP_2:68</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:126"/>
<font color="Maroon">H</font> is  <a href="group_3.html#V1" target="_self">normal</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>
 ;<br/>


</div>
