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

<b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/>

<a NAME="E1:28"/><i><font color="Green">E23</font></i>: 
 <a href="group_4.html#K11">lattice</a> <font color="Maroon">G</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">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K9">SubJoin</a> <font color="Maroon">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K10">SubMeet</a> <font color="Maroon">G</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="group_4.html#D13">GROUP_4:def 13</a></i>;<br/>
<b>let </b><font color="Maroon">Y</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="group_4.html#K11">lattice</a> <font color="Maroon">G</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_8.html#D6">VECTSP_8:def 6</a><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1"/>
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1"><b>suppose </b></a><a NAME="E1:28_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>take </b>
 <a href="lattices.html#K6">Top</a> <span class="p1">(<span class="default"><a href="group_4.html#K11">lattice</a> <font color="Maroon">G</font></span>)</span>
;<br/><b>thus </b><a NAME="E2:28_1_1"/>
 <a href="lattices.html#K6">Top</a> <span class="p1">(<span class="default"><a href="group_4.html#K11">lattice</a> <font color="Maroon">G</font></span>)</span> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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">G</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a><br/>

<b>thus </b><a NAME="E1:28_1_1_1"/>
(  not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> or  <a href="lattices.html#K6">Top</a> <span class="p1">(<span class="default"><a href="group_4.html#K11">lattice</a> <font color="Maroon">G</font></span>)</span> <a href="lattices.html#R1">[=</a> <font color="Maroon">q</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon">b</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">G</font></span>)</span>;<br/><b>assume </b><a NAME="E3:28_1_1"/>
<font color="Maroon">b</font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">Y</font>
 ;<br/><b>thus </b><a NAME="E4:28_1_1"/>
<font color="Maroon">b</font> <a href="lattices.html#R1">[=</a>  <a href="lattices.html#K6">Top</a> <span class="p1">(<span class="default"><a href="group_4.html#K11">lattice</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="lattices.html#T45">LATTICES:45</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_2"><b>suppose </b></a><a NAME="E1:28_1_2"/>
<font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">Y</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="group_3.html#K1">Subgroups</a> <font color="Maroon">G</font></span>)</span> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">p</font> =  <a href="latsubgr.html#K2" target="_self">meet</a> <font color="Maroon">X</font> as   <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">G</font></span>)</span> <b>by </b><i>, <a class="ref" href="group_3.html#D1">GROUP_3:def 1</a></i>;<br/><b>take </b>
<font color="Maroon">p</font>
;<br/><b>thus </b><a NAME="E4:28_1_2"/>
<font color="Maroon">p</font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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">G</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a><br/>

<b>reconsider </b><font color="Maroon">H</font> = <font color="Maroon">q</font> as  <a href="group_1.html#V1">strict</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#D1">GROUP_3:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">q</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="group_3.html#K1">Subgroups</a> <font color="Maroon">G</font> <b>by </b><i/>;<br/>
<a NAME="E3:28_1_2_1"/><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font>
 
<b>by </b><i/>;<br/>
<b>assume </b><a NAME="E4:28_1_2_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E5:28_1_2_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E6:28_1_2_1"/><b>then </b>
 <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/>
<a NAME="E7:28_1_2_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="latsubgr.html#K2" target="_self">meet</a> <font color="Maroon">X</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:28_1_2_1"/>
<font color="Maroon">p</font> <a href="lattices.html#R1">[=</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="latsubgr.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon">r</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">G</font></span>)</span>;<br/><b>assume </b><a NAME="E5:28_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">r</font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">Y</font>
 ;<br/><b>reconsider </b><font color="Maroon">H</font> = <font color="Maroon">r</font> as    <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#D1">GROUP_3:def 1</a></i>;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">X</font>;<br/><a NAME="E8:28_1_2"/><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/><a NAME="E9:28_1_2"/>
for <font color="Olive">Z1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> holds <br/>the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Z1</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:28_1_2_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">Z2</font> = <font color="Maroon">Z1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> ;<br/>
<b>consider </b><font color="Maroon">z1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="group_3.html#K1">Subgroups</a> <font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:28_1_2_2"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">Z2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z1</font> )
 <b>by </b><i><a class="ref" href="latsubgr.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>reconsider </b><font color="Maroon">z1</font> = <font color="Maroon">z1</font> as   <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">G</font></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">z3</font> = <font color="Maroon">z1</font> as  <a href="group_1.html#V1">strict</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#D1">GROUP_3:def 1</a></i>;<br/>
<a NAME="E7:28_1_2_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">Z1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">z3</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:28_1_2_2"/>
<font color="Maroon">r</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">z1</font>
 
<b>by </b><i>, , <a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/>
<b>hence </b><a NAME="E9:28_1_2_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z1</font>
 <b>by </b><i>, <a class="ref" href="latsubgr.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:28_1_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K1" target="_self">carr</a> <font color="Maroon">G</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<br/><a NAME="E11:28_1_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="latsubgr.html#K2" target="_self">meet</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E12:28_1_2"/>
<font color="Maroon">r</font> <a href="lattices.html#R1">[=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="latsubgr.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
