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<b>let </b><font color="Maroon" title="c1">K</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>;<br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">K</font>;<br/>



<b>set </b><font color="Maroon" title="c3">S0</font> =  <a href="lmod_7.html#K1" title="LMOD_7:func.1">Submodules</a> <font color="Maroon" title="c2">V</font>;<br/>
<b>set </b><font color="Maroon" title="c4">S1</font> =  <a href="vectsp_5.html#K5" title="VECTSP_5:func.5">SubJoin</a> <font color="Maroon" title="c2">V</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">L</font> =  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="lmod_7.html#K1" title="LMOD_7:func.1">Submodules</a> <font color="Maroon" title="c2">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp_5.html#K5" title="VECTSP_5:func.5">SubJoin</a> <font color="Maroon" title="c2">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp_5.html#K6" title="VECTSP_5:func.6">SubMeet</a> <font color="Maroon" title="c2">V</font></span>)</span> #) as   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a> <b>by </b><i><a class="ref" href="vectsp_5.html#T75" title="VECTSP_5:th.75">VECTSP_5:75</a></i>;<br/>
<a NAME="E2:31"/>
 <a href="vectsp_5.html#K5" title="VECTSP_5:func.5">SubJoin</a> <font color="Maroon" title="c2">V</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c5">L</font>
 
;<br/>
<b>hence </b><a NAME="E3:31"/>
(  <a href="vectsp_5.html#K5" title="VECTSP_5:func.5">SubJoin</a> <font color="Maroon" title="c2">V</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp;  <a href="vectsp_5.html#K5" title="VECTSP_5:func.5">SubJoin</a> <font color="Maroon" title="c2">V</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> )
 ;<br/>

<b>set </b><font color="Maroon" title="c6">e</font> =  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">e'</font> =  <a href="lmod_6.html#K1" title="LMOD_6:func.1">@</a> <span class="p1">(<span class="default"><a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lmod_7.html#K1" title="LMOD_7:func.1">Submodules</a> <font color="Maroon" title="c2">V</font> ;<br/>
<a NAME="E5:31"/><i><font color="Green" title="E10">A1</font></i>: 
<font color="Maroon" title="c7">e'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font>
 
<b>by </b><i><a class="ref" href="lmod_6.html#D3" title="LMOD_6:def.3">LMOD_6:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">a'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lmod_7.html#K1" title="LMOD_7:func.1">Submodules</a> <font color="Maroon" title="c2">V</font>;<br/><b>reconsider </b><font color="Maroon" title="c9">b</font> = <font color="Maroon" title="c8">a'</font> as    <a href="lmod_7.html#M2" title="LMOD_7:mode.2">Element</a> of  <a href="lmod_7.html#K1" title="LMOD_7:func.1">Submodules</a> <font color="Maroon" title="c2">V</font> ;<br/><b>reconsider </b><font color="Maroon" title="c10">a</font> = <font color="Maroon" title="c9">b</font> as  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font> ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="vectsp_5.html#K5" title="VECTSP_5:func.5">SubJoin</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">e'</font>,<font color="Maroon" title="c8">a'</font> = 
<span class="p1">(<span class="default"><a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c10">a</font>
<b>by </b><i><a class="txt" href="lmod_7.html#E5:31"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="vectsp_5.html#D7" title="VECTSP_5:def.7">VECTSP_5:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c8">a'</font>
<b>by </b><i><a class="ref" href="vectsp_5.html#T13" title="VECTSP_5:th.13">VECTSP_5:13</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><a href="vectsp_5.html#K5" title="VECTSP_5:func.5">SubJoin</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c7">e'</font> = 
<font color="Maroon" title="c10">a</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font></span>)</span>
<b>by </b><i><a class="txt" href="lmod_7.html#E5:31"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="vectsp_5.html#D7" title="VECTSP_5:def.7">VECTSP_5:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c8">a'</font>
<b>by </b><i><a class="ref" href="vectsp_5.html#T13" title="VECTSP_5:th.13">VECTSP_5:13</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E7:31"/><b>then </b><i><font color="Green" title="E11">A2</font></i>: 
<font color="Maroon" title="c7">e'</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a>  <a href="vectsp_5.html#K5" title="VECTSP_5:func.5">SubJoin</a> <font color="Maroon" title="c2">V</font>
 
<b>by </b><i><a class="ref" href="binop_1.html#T11" title="BINOP_1:th.11">BINOP_1:11</a></i>;<br/>
<b>hence </b><a NAME="E8:31"/>
 <a href="vectsp_5.html#K5" title="VECTSP_5:func.5">SubJoin</a> <font color="Maroon" title="c2">V</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a> 
 <b>by </b><i><a class="ref" href="setwiseo.html#D2" title="SETWISEO:def.2">SETWISEO:def 2</a></i>;<br/>

<b>thus </b><a NAME="E9:31"/>
 <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <span class="p1">(<span class="default"><a href="vectsp_5.html#K5" title="VECTSP_5:func.5">SubJoin</a> <font color="Maroon" title="c2">V</font></span>)</span>
 <b>by </b><i><a class="txt" href="lmod_7.html#E5:31"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="lmod_7.html#E7:31"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="binop_1.html#D8" title="BINOP_1:def.8">BINOP_1:def 8</a></i>;<br/>


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