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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>thus </b><a NAME="E1:128"/>
( <font color="Maroon">c<sub>2</sub></font> is  <a href="group_3.html#V1" target="_self">normal</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font> implies for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K13">*</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="group_3.html#T140" target="_self">Th140</a></i>;<br/>

<b>assume </b><a NAME="E2:128"/><i><font color="Green">E90</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K13">*</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><i><font color="Green">E91</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="128_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:128_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K13">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="group_3.html#E2:128"><i><font color="Green">E90</font></i></a></i>;<br/><a NAME="E2:128_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K13">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="group_3.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E3:128_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K13">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="group_2.html#T127">GROUP_2:127</a></i>;<br/><a NAME="E4:128_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K13">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>;<br/><a NAME="E5:128_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K13">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="group_2.html#T132">GROUP_2:132</a></i>;<br/><a NAME="E6:128_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="group_2.html#T128">GROUP_2:128</a></i>;<br/><a NAME="E7:128_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="group_3.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E8:128_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E9:128_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="group_2.html#T40">GROUP_2:40</a></i>;<br/><a NAME="E10:128_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K4">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>;<br/><b>hence </b><a NAME="E11:128_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="group_2.html#T43">GROUP_2:43</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="128_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:128_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="group_3.html#E2:128"><i><font color="Green">E90</font></i></a>, <a class="txt" href="group_3.html#E3:128"><i><font color="Green">E91</font></i></a></i>;<br/><b>hence </b><a NAME="E2:128_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:128"/>
<font color="Maroon">c<sub>2</sub></font> is  <a href="group_3.html#V1" target="_self">normal</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="group_3.html#T140" target="_self">Th140</a></i>;<br/>


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