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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">H</font> being  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> holds <br/> ( <font color="Olive" title="b1">H</font> is   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> iff  <a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">H</font> be   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">H</font> is   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> iff  <a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> )</span><br/>



<span class="kw">thus </span><a NAME="E1:172"/>
( <font color="Maroon" title="c2">H</font> is   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> implies  <a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> implies <font color="Maroon" title="c2">H</font> is   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="172_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:172_1"/><span class="lab"><font color="Green" title="E93">A1</font></span>: 
<font color="Maroon" title="c2">H</font> is   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="172_1_1"><span class="kw">now </span></a><div class="add"><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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c2">H</font></span><br/><a NAME="E1:172_1_1"/>
<font color="Maroon" title="c2">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c3">a</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:172_1"><span class="lab"><font color="Green" title="E93">A1</font></span></a>, <a class="ref" href="group_3.html#D13" target="_self" title="GROUP_3:def.13">Def13</a></span>;<br/><span class="kw">hence </span><a NAME="E2:172_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T160" target="_self" title="GROUP_3:th.160">Th160</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:172_1"/>
 <a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T71" title="GROUP_2:th.71">GROUP_2:71</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:172"/><span class="lab"><font color="Green" title="E93">A2</font></span>: 
 <a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">H</font> is   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font></span><br/>

<a NAME="E3:172"/>
<font color="Maroon" title="c2">H</font> is  <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="172_2"><span class="kw">proof </span></a><div class="add">

<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>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="group_3.html#D13" target="_self" title="GROUP_3:def.13">GROUP_3:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">H</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">H</font> #)</span><br/>

<a NAME="E1:172_2"/>
<font color="Maroon" title="c3">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c2">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:172"><span class="lab"><font color="Green" title="E93">A2</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E2:172_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c2">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Olive" title="b1">b</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c2">H</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T160" target="_self" title="GROUP_3:th.160">Th160</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:172_2"/>
<font color="Maroon" title="c2">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">H</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">H</font> #)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:172"/>
<font color="Maroon" title="c2">H</font> is   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
