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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="vectsp_2.html#NM6">LeftMod</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="lmod_7.html#K1" target="_self">Submodules</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="vectsp_5.html#K5">SubJoin</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="lmod_7.html#K1" target="_self">Submodules</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp_5.html#K5">SubJoin</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp_5.html#K6">SubMeet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #) as   <a href="lattices.html#NM1">Lattice</a> <b>by </b><i><a class="ref" href="vectsp_5.html#T75">VECTSP_5:75</a></i>;<br/>
<a NAME="E2:31"/>
 <a href="vectsp_5.html#K5">SubJoin</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R1">=</a> the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>5</sub></font>
 
;<br/>
<b>hence </b><a NAME="E3:31"/>
(  <a href="vectsp_5.html#K5">SubJoin</a> <font color="Maroon">c<sub>2</sub></font> is <a href="binop_1.html#V1">commutative</a> &amp;  <a href="vectsp_5.html#K5">SubJoin</a> <font color="Maroon">c<sub>2</sub></font> is <a href="binop_1.html#V2">associative</a> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="lmod_6.html#K1">@</a> <span class="p1">(<span class="default"><a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="lmod_7.html#K1" target="_self">Submodules</a> <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E5:31"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="lmod_6.html#D3">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">c<sub>8</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lmod_7.html#K1" target="_self">Submodules</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="lmod_7.html#M2" target="_self">Element</a> of  <a href="lmod_7.html#K1" target="_self">Submodules</a> <font color="Maroon">c<sub>2</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="vectsp_5.html#K5">SubJoin</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="vectsp_5.html#K1">+</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="lmod_7.html#E5:31"><i><font color="Green">E10</font></i></a>, <a class="ref" href="vectsp_5.html#D7">VECTSP_5:def 7</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="vectsp_5.html#T13">VECTSP_5:13</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><a href="vectsp_5.html#K5">SubJoin</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>10</sub></font> <a href="vectsp_5.html#K1">+</a> <span class="p1">(<span class="default"><a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="lmod_7.html#E5:31"><i><font color="Green">E10</font></i></a>, <a class="ref" href="vectsp_5.html#D7">VECTSP_5:def 7</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="vectsp_5.html#T13">VECTSP_5:13</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E7:31"/><b>then </b><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="binop_1.html#R3">is_a_unity_wrt</a>  <a href="vectsp_5.html#K5">SubJoin</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="binop_1.html#T11">BINOP_1:11</a></i>;<br/>
<b>hence </b><a NAME="E8:31"/>
 <a href="vectsp_5.html#K5">SubJoin</a> <font color="Maroon">c<sub>2</sub></font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a> 
 <b>by </b><i><a class="ref" href="setwiseo.html#D2">SETWISEO:def 2</a></i>;<br/>

<b>thus </b><a NAME="E9:31"/>
 <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <span class="p1">(<span class="default"><a href="vectsp_5.html#K5">SubJoin</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="lmod_7.html#E5:31"><i><font color="Green">E10</font></i></a>, <a class="txt" href="lmod_7.html#E7:31"><i><font color="Green">E11</font></i></a>, <a class="ref" href="binop_1.html#D8">BINOP_1:def 8</a></i>;<br/>


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