<?xml version="1.0"?>
<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_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>thus </b><a NAME="E1:130"/>
( <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> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>2</sub></font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Olive">b<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="130_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:130_1"/><i><font color="Green">E90</font></i>: 
<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>
 ;<br/>

<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="E2:130_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font>,the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>2</sub></font> #)
 
<b>by </b><i><a class="txt" href="group_3.html#E1:130_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="group_3.html#D13" target="_self">Def13</a></i>;<br/>
<b>hence </b><a NAME="E3:130_1"/>
<font color="Maroon">c<sub>2</sub></font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>2</sub></font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="group_2.html#T63">GROUP_2:63</a></i>;<br/>


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

<b>assume </b><a NAME="E2:130"/><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> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>2</sub></font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="130_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="E1:130_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>2</sub></font> <a href="group_3.html#K6" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="group_3.html#E2:130"><i><font color="Green">E90</font></i></a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_3.html#K6" target="_self">|^</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#K13">*</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_3.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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#K3">"</a> </span>)</span> <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>
<b>by </b><i><a class="ref" href="group_3.html#T71" target="_self">Th71</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K3">"</a> </span>)</span> <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>
<b>by </b><i><a class="ref" href="group_2.html#T40">GROUP_2:40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K3">"</a> </span>)</span> <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>
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>
<br/>.= 
<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#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="ref" href="group_2.html#T43">GROUP_2:43</a></i>
<br/>.= 
<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="ref" href="group_1.html#T19">GROUP_1:19</a></i>
;<br/><b>hence </b><a NAME="E3:130_2"/>
<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#E1:130_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="group_3.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:130"/>
<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#T142" target="_self">Th142</a></i>;<br/>


</div>
