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<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>, <font color="Maroon">c<sub>3</sub></font> be    <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>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="group_4.html#K11">lattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:27"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E2:27"/><b>then </b><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is  <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> &amp; <font color="Maroon">c<sub>3</sub></font> is  <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> )
 
<b>by </b><i><a class="ref" href="latsubgr.html#T29" target="_self">Th29</a></i>;<br/>
<b>thus </b><a NAME="E3:27"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>5</sub></font> implies <font color="Maroon">c<sub>2</sub></font> is    <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:27_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="latsubgr.html#E1:27"><i><font color="Green">E15</font></i></a>, <a class="ref" href="latsubgr.html#T30" target="_self">Th30</a></i>;<br/>
<b>hence </b><a NAME="E3:27_1"/>
<font color="Maroon">c<sub>2</sub></font> is    <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="group_2.html#T66">GROUP_2:66</a></i>;<br/>


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

<b>thus </b><a NAME="E4:27"/>
( <font color="Maroon">c<sub>2</sub></font> is    <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>5</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_2"/>
<font color="Maroon">c<sub>2</sub></font> is    <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:27_2"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="latsubgr.html#E2:27"><i><font color="Green">E16</font></i></a>, <a class="ref" href="group_2.html#T107">GROUP_2:107</a></i>;<br/>
<a NAME="E3:27_2"/>
 <a href="group_4.html#K11">lattice</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="group_3.html#K1">Subgroups</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K9">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K10">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="group_4.html#D13">GROUP_4:def 13</a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> = 
the <a href="lattices.html#U1">L_meet</a> of <span class="p1">(<span class="default"><a href="group_4.html#K11">lattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="latsubgr.html#E1:27"><i><font color="Green">E15</font></i></a>, <a class="ref" href="group_4.html#D12">GROUP_4:def 12</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
;<br/>
<b>hence </b><a NAME="E5:27_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="latsubgr.html#E1:27"><i><font color="Green">E15</font></i></a>, <a class="txt" href="latsubgr.html#E2:27_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="lattices.html#T21">LATTICES:21</a></i>;<br/>


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


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