<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">H2</font> being    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font><br/>  for <font color="Olive" title="b2">H1</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="b2">H1</font>,<font color="Olive" title="b1">H2</font> <a href="group_3.html#R5" title="GROUP_3:pred.5">are_conjugated</a>  iff  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Olive" title="b2">H1</font>, <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Olive" title="b1">H2</font> <a href="group_3.html#R4" title="GROUP_3:pred.4">are_conjugated</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">H2</font> be    <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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">H1</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">H1</font>,<font color="Maroon" title="c2">H2</font> <a href="group_3.html#R5" title="GROUP_3:pred.5">are_conjugated</a>  iff  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Olive" title="b1">H1</font>, <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H2</font> <a href="group_3.html#R4" title="GROUP_3:pred.4">are_conjugated</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">H1</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">H1</font>,<font color="Maroon" title="c2">H2</font> <a href="group_3.html#R5" title="GROUP_3:pred.5">are_conjugated</a>  iff  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H1</font>, <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H2</font> <a href="group_3.html#R4" title="GROUP_3:pred.4">are_conjugated</a>  )</span><br/>





<span class="kw">thus </span><a NAME="E1:117"/>
( <font color="Maroon" title="c3">H1</font>,<font color="Maroon" title="c2">H2</font> <a href="group_3.html#R5" title="GROUP_3:pred.5">are_conjugated</a>  implies  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H1</font>, <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H2</font> <a href="group_3.html#R4" title="GROUP_3:pred.4">are_conjugated</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H1</font>, <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H2</font> <a href="group_3.html#R4" title="GROUP_3:pred.4">are_conjugated</a>  implies <font color="Maroon" title="c3">H1</font>,<font color="Maroon" title="c2">H2</font> <a href="group_3.html#R5" title="GROUP_3:pred.5">are_conjugated</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c4">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><a NAME="E2:117_1"/><span class="lab"><font color="Green" title="E79">A1</font></span>: 
 <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="c3">H1</font>, the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">H1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H2</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c4">a</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="group_3.html#D11" target="_self" title="GROUP_3:def.11">GROUP_3:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H1</font>, <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H2</font> <a href="group_3.html#R4" title="GROUP_3:pred.4">are_conjugated</a> </span><br/>

<a NAME="E3:117_1"/>
 <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H2</font></span>)</span> <a href="group_3.html#K4" title="GROUP_3:func.4">|^</a> <font color="Maroon" title="c4">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:117_1"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="ref" href="group_3.html#D6" target="_self" title="GROUP_3:def.6">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:117_1"/>
 <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H1</font>, <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H2</font> <a href="group_3.html#R4" title="GROUP_3:pred.4">are_conjugated</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#D9" target="_self" title="GROUP_3:def.9">Def9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">given </span><font color="Maroon" title="c4">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><a NAME="E3:117"/><span class="lab"><font color="Green" title="E79">A2</font></span>: 
 <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H2</font></span>)</span> <a href="group_3.html#K4" title="GROUP_3:func.4">|^</a> <font color="Maroon" title="c4">a</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="group_3.html#D9" target="_self" title="GROUP_3:def.9">GROUP_3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">H1</font>,<font color="Maroon" title="c2">H2</font> <a href="group_3.html#R5" title="GROUP_3:pred.5">are_conjugated</a> </span><br/>

<a NAME="E4:117"/>
<font color="Maroon" title="c3">H1</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c2">H2</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c4">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:117"><span class="lab"><font color="Green" title="E79">A2</font></span></a>, <a class="ref" href="group_3.html#D6" target="_self" title="GROUP_3:def.6">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:117"/>
<font color="Maroon" title="c3">H1</font>,<font color="Maroon" title="c2">H2</font> <a href="group_3.html#R5" title="GROUP_3:pred.5">are_conjugated</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#D11" target="_self" title="GROUP_3:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
