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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c2">H1</font>, <font color="Maroon" title="c3">H2</font> be    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>;<br/><b>let </b><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="group_4.html#K11" title="GROUP_4:func.11">lattice</a> <font color="Maroon" title="c1">G</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:26"/><i><font color="Green" title="E15">A1</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H1</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H2</font> )
 ;<br/>

<a NAME="E2:26"/><i><font color="Green" title="E16">A2</font></i>: 
 <a href="group_4.html#K11" title="GROUP_4:func.11">lattice</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K9" title="GROUP_4:func.9">SubJoin</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K10" title="GROUP_4:func.10">SubMeet</a> <font color="Maroon" title="c1">G</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="group_4.html#D13" title="GROUP_4:def.13">GROUP_4:def 13</a></i>;<br/>
<a NAME="E3:26"/><i><font color="Green" title="E17">A3</font></i>: 
( <font color="Maroon" title="c2">H1</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c3">H2</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> )
 
<b>by </b><i><a class="txt" href="latsubgr.html#E1:26"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="latsubgr.html#T29" target="_self" title="LATSUBGR:th.29">Th29</a></i>;<br/>
<b>thus </b><a NAME="E4:26"/>
( <font color="Maroon" title="c4">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">q</font> implies the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">H1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_1"/>
<font color="Maroon" title="c4">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">q</font>
 ;<br/>

<a NAME="E2:26_1"/><b>then </b><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T21" title="LATTICES:th.21">LATTICES:21</a></i>;<br/>
<font color="Maroon" title="c4">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">q</font> = 
the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <span class="p1">(<span class="default"><a href="group_4.html#K11" title="GROUP_4:func.11">lattice</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>
<b>by </b><i><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c2">H1</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c3">H2</font>
<b>by </b><i><a class="txt" href="latsubgr.html#E1:26"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="latsubgr.html#E2:26"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="group_4.html#D12" title="GROUP_4:def.12">GROUP_4:def 12</a></i>
;<br/>
<a NAME="E4:26_1"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">H1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">H1</font>
 
<b>by </b><i><a class="txt" href="latsubgr.html#E1:26"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="latsubgr.html#E2:26_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="latsubgr.html#T1" target="_self" title="LATSUBGR:th.1">Th1</a></i>;<br/>
<b>hence </b><a NAME="E5:26_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">H1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>


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

<b>thus </b><a NAME="E5:26"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">H1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H2</font> implies <font color="Maroon" title="c4">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">q</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">H1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H2</font>
 ;<br/>

<a NAME="E2:26_2"/><b>then </b>
<font color="Maroon" title="c2">H1</font> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c3">H2</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T66" title="GROUP_2:th.66">GROUP_2:66</a></i>;<br/>
<a NAME="E3:26_2"/><b>then </b><i><font color="Green" title="E18">A5</font></i>: 
<font color="Maroon" title="c2">H1</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c3">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H1</font>
 
<b>by </b><i><a class="txt" href="latsubgr.html#E3:26"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="group_2.html#T107" title="GROUP_2:th.107">GROUP_2:107</a></i>;<br/>
<font color="Maroon" title="c2">H1</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c3">H2</font> = 
the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <span class="p1">(<span class="default"><a href="group_4.html#K11" title="GROUP_4:func.11">lattice</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>
<b>by </b><i><a class="txt" href="latsubgr.html#E1:26"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="latsubgr.html#E2:26"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="group_4.html#D12" title="GROUP_4:def.12">GROUP_4:def 12</a></i>
<br/>.= 
<font color="Maroon" title="c4">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">q</font>
<b>by </b><i><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
;<br/>
<b>hence </b><a NAME="E5:26_2"/>
<font color="Maroon" title="c4">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">q</font>
 <b>by </b><i><a class="txt" href="latsubgr.html#E1:26"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="latsubgr.html#E3:26_2"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="lattices.html#T21" title="LATTICES:th.21">LATTICES:21</a></i>;<br/>


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


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