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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">H</font>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>] </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> <b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<a NAME="E2:55_1_1"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">H</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:55_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
;<br/>
<div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1"><b>now </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>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:55_1_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E2:55_1_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E3:55_1_1_1"/><i><font color="Green">E53</font></i>: 
 ex <font color="Olive">a1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a1</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">H</font> )
 <b>by </b><i/>;<br/><a NAME="E4:55_1_1_1"/><i><font color="Green">E73</font></i>: 
 ex <font color="Olive">b1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b1</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">H</font> )
 <b>by </b><i/>;<br/><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">H</font>
<b>by </b><i><a class="txt" href="group_6.html#E3:55_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="group_6.html#E4:55_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/><b>hence </b><a NAME="E6:55_1_1_1"/>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_2"><b>now </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>;<br/><b>assume </b><a NAME="E1:55_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:55_1_1_2"/><b>then </b>
 ex <font color="Olive">a1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a1</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">H</font> )
 ;<br/><b>then </b><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">H</font></span>)</span> <a href="group_1.html#K3">"</a> 
<b>by </b><i/>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">H</font>
<b>by </b><i><a class="ref" href="group_1.html#T16">GROUP_1:16</a></i>
;<br/><b>hence </b><a NAME="E4:55_1_1_2"/>
<font color="Maroon">a</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">B</font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E7:55_1_1"/><i><font color="Green">E74</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="group_2.html#T61">GROUP_2:61</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_3"><b>now </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>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E1:55_1_1_3_1"/>
<font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">B</font> <a href="group_3.html#K6">|^</a> <font color="Maroon">a</font>
 ;<br/><b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:55_1_1_3_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="group_3.html#K2">|^</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E4:55_1_1_3_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">c</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="group_3.html#T70">GROUP_3:70</a></i>;<br/><a NAME="E5:55_1_1_3_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="txt" href="group_6.html#E3:55_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E6:55_1_1_3_1"/><b>then </b>
 ex <font color="Olive">a1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">a1</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a1</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">H</font> )
 ;<br/><b>then </b><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">H</font></span>)</span> <a href="group_3.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">H</font>
<b>by </b><i><a class="ref" href="group_3.html#T22">GROUP_3:22</a></i>
;<br/><a NAME="E8:55_1_1_3_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>hence </b><a NAME="E9:55_1_1_3_1"/>
<font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:55_1_1_3"/>
<font color="Maroon">B</font> <a href="group_3.html#K6">|^</a> <font color="Maroon">a</font> is    <a href="group_6.html#M1" target="_self">Subgroup</a> of <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="group_2.html#T67">GROUP_2:67</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:55_1_1"/><b>then </b>
<font color="Maroon">B</font> is  <a href="group_3.html#V1">normal</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T145">GROUP_3:145</a></i>;<br/>
<b>hence </b><a NAME="E10:55_1_1"/>
 <a href="group_6.html#K11" target="_self">Ker</a> <font color="Maroon">g</font> is <a href="group_3.html#V1">normal</a>
 <b>by </b><i>, </i>;<br/>


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