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

<b>let </b><font color="Maroon">R</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_2.html#NM7">RightMod</a> of <font color="Maroon">R</font>;<br/>



<b>set </b><font color="Maroon">S</font> =  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rmod_3.html#K3" target="_self">Submodules</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K5" target="_self">SubJoin</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K6" target="_self">SubMeet</a> <font color="Maroon">V</font></span>)</span> #);<br/>
<a NAME="E1:68"/>
for <font color="Olive">A</font>, <font color="Olive">B</font>, <font color="Olive">C</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rmod_3.html#K3" target="_self">Submodules</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K5" target="_self">SubJoin</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K6" target="_self">SubMeet</a> <font color="Maroon">V</font></span>)</span> #)  st <font color="Olive">A</font> <a href="lattices.html#R1">[=</a> <font color="Olive">C</font> holds <br/><font color="Olive">A</font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Olive">B</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">C</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">A</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">B</font></span>)</span> <a href="lattices.html#K2">"/\"</a> <font color="Olive">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rmod_3.html#K3" target="_self">Submodules</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K5" target="_self">SubJoin</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K6" target="_self">SubMeet</a> <font color="Maroon">V</font></span>)</span> #), <font color="Maroon">B</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rmod_3.html#K3" target="_self">Submodules</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K5" target="_self">SubJoin</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K6" target="_self">SubMeet</a> <font color="Maroon">V</font></span>)</span> #), <font color="Maroon">C</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rmod_3.html#K3" target="_self">Submodules</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K5" target="_self">SubJoin</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K6" target="_self">SubMeet</a> <font color="Maroon">V</font></span>)</span> #);<br/>

<b>assume </b><a NAME="E1:68_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">A</font> <a href="lattices.html#R1">[=</a> <font color="Maroon">C</font>
 ;<br/>

<b>consider </b><font color="Maroon">W1</font> being  <a href="vectsp_2.html#V3">strict</a>  <a href="rmod_2.html#M1">Submodule</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:68_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="rmod_3.html#E3:68_1"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">W2</font> being  <a href="vectsp_2.html#V3">strict</a>  <a href="rmod_2.html#M1">Submodule</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E5:68_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">W2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="rmod_3.html#E3:68_1"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">W3</font> being  <a href="vectsp_2.html#V3">strict</a>  <a href="rmod_2.html#M1">Submodule</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E7:68_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">W3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="rmod_3.html#E3:68_1"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">BC</font> = <font color="Maroon">W2</font> <a href="rmod_3.html#K2" target="_self">/\</a> <font color="Maroon">W3</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rmod_3.html#K3" target="_self">Submodules</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K5" target="_self">SubJoin</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K6" target="_self">SubMeet</a> <font color="Maroon">V</font></span>)</span> #) <b>by </b><i><a class="txt" href="rmod_3.html#E3:68_1"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">AB</font> = <font color="Maroon">W1</font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W2</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rmod_3.html#K3" target="_self">Submodules</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K5" target="_self">SubJoin</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K6" target="_self">SubMeet</a> <font color="Maroon">V</font></span>)</span> #) <b>by </b><i><a class="txt" href="rmod_3.html#E3:68_1"><i><font color="Green">E33</font></i></a></i>;<br/>
<font color="Maroon">W1</font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W3</font> = 
<span class="p1">(<span class="default"><a href="rmod_3.html#K5" target="_self">SubJoin</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">A</font>,<font color="Maroon">C</font>
<b>by </b><i><a class="txt" href="rmod_3.html#E11"><i><font color="Green">Lemma48</font></i></a>, <a class="ref" href="rmod_3.html#T10" target="_self">Th10</a>, </i>
<br/>.= 
<font color="Maroon">A</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">C</font>
<b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
<br/>.= 
<font color="Maroon">W3</font>
<b>by </b><i>, <a class="ref" href="rmod_3.html#T10" target="_self">Th10</a>, <a class="ref" href="lattices.html#D3">LATTICES:def 3</a></i>
;<br/>
<a NAME="E11:68_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">W1</font> is    <a href="rmod_2.html#M1">Submodule</a> of <font color="Maroon">W3</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><font color="Maroon">A</font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">C</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rmod_3.html#K5" target="_self">SubJoin</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">C</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rmod_3.html#K5" target="_self">SubJoin</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rmod_3.html#K6" target="_self">SubMeet</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rmod_3.html#K5" target="_self">SubJoin</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">A</font>,<font color="Maroon">BC</font>
<b>by </b><i><a class="txt" href="rmod_3.html#E12"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="rmod_3.html#T10" target="_self">Th10</a>, <a class="ref" href="rmod_3.html#T5" target="_self">Th5</a></i>
<br/>.= 
<font color="Maroon">W1</font> <a href="rmod_3.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="rmod_3.html#K2" target="_self">/\</a> <font color="Maroon">W3</font></span>)</span>
<b>by </b><i><a class="txt" href="rmod_3.html#E11"><i><font color="Green">Lemma48</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="rmod_3.html#K1" target="_self">+</a> <font color="Maroon">W2</font></span>)</span> <a href="rmod_3.html#K2" target="_self">/\</a> <font color="Maroon">W3</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rmod_3.html#K6" target="_self">SubMeet</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">AB</font>,<font color="Maroon">C</font>
<b>by </b><i><a class="ref" href="rmod_3.html#T10" target="_self">Th10</a>, <a class="ref" href="rmod_3.html#T5" target="_self">Th5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rmod_3.html#K6" target="_self">SubMeet</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rmod_3.html#K5" target="_self">SubJoin</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">C</font>
<b>by </b><i><a class="txt" href="rmod_3.html#E11"><i><font color="Green">Lemma48</font></i></a>, <a class="txt" href="rmod_3.html#E12"><i><font color="Green">Lemma49</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rmod_3.html#K6" target="_self">SubMeet</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">C</font>
<b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">B</font></span>)</span> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">C</font>
<b>by </b><i><a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:68"/>
 <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="rmod_3.html#K3" target="_self">Submodules</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K5" target="_self">SubJoin</a> <font color="Maroon">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="rmod_3.html#K6" target="_self">SubMeet</a> <font color="Maroon">V</font></span>)</span> #) is   <a href="lattices.html#NM3">M_Lattice</a>
 <b>by </b><i><a class="ref" href="rmod_3.html#T7" target="_self">Th7</a>, <a class="ref" href="lattices.html#D12">LATTICES:def 12</a></i>;<br/>


</div>
