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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>let </b><font color="Maroon" title="c3">W</font> be    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font>;<br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Scalar</a> of <font color="Maroon" title="c1">K</font>;<br/><b>let </b><font color="Maroon" title="c6">v</font>, <font color="Maroon" title="c7">w</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Maroon" title="c3">W</font></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c8">a</font> being   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E2:58_1"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Maroon" title="c3">W</font>
 <b>by </b><i><a class="ref" href="lmod_7.html#T28" target="_self" title="LMOD_7:th.28">Th28</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">b</font> being   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E4:58_1"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Maroon" title="c3">W</font>
 <b>by </b><i><a class="ref" href="lmod_7.html#T28" target="_self" title="LMOD_7:th.28">Th28</a></i>;<br/><a NAME="E5:58_1"/><i><font color="Green" title="E29">A3</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c6">v</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c7">w</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c6">v</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c7">w</font> )
 <b>by </b><i><a class="txt" href="lmod_7.html#E2:58_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="lmod_7.html#E4:58_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="lmod_7.html#T30" target="_self" title="LMOD_7:th.30">Th30</a></i>;<br/><b>thus </b><font color="Maroon" title="c4">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c7">w</font></span>)</span> = 
<font color="Maroon" title="c4">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Maroon" title="c3">W</font></span>)</span>
<b>by </b><i><a class="txt" href="lmod_7.html#E2:58_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="lmod_7.html#E4:58_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="lmod_7.html#T30" target="_self" title="LMOD_7:th.30">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">b</font></span>)</span></span>)</span> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Maroon" title="c3">W</font>
<b>by </b><i><a class="ref" href="lmod_7.html#T30" target="_self" title="LMOD_7:th.30">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c9">b</font></span>)</span></span>)</span> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Maroon" title="c3">W</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c6">v</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c7">w</font></span>)</span>
<b>by </b><i><a class="txt" href="lmod_7.html#E5:58_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="lmod_7.html#T30" target="_self" title="LMOD_7:th.30">Th30</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c6">v</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Maroon" title="c3">W</font>
<b>by </b><i><a class="txt" href="lmod_7.html#E2:58_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="lmod_7.html#T30" target="_self" title="LMOD_7:th.30">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Maroon" title="c3">W</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c6">v</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c6">v</font></span>)</span>
<b>by </b><i><a class="txt" href="lmod_7.html#E5:58_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="lmod_7.html#T30" target="_self" title="LMOD_7:th.30">Th30</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c6">v</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Maroon" title="c3">W</font>
<b>by </b><i><a class="txt" href="lmod_7.html#E2:58_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="lmod_7.html#T30" target="_self" title="LMOD_7:th.30">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Maroon" title="c3">W</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>
<br/>.= 
<font color="Maroon" title="c4">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Maroon" title="c3">W</font></span>)</span>
<b>by </b><i><a class="ref" href="lmod_7.html#T30" target="_self" title="LMOD_7:th.30">Th30</a></i>
<br/>.= 
<font color="Maroon" title="c4">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c6">v</font></span>)</span>
<b>by </b><i><a class="txt" href="lmod_7.html#E2:58_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="lmod_7.html#T30" target="_self" title="LMOD_7:th.30">Th30</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c6">v</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="lmod_7.html#K18" title="LMOD_7:func.18">/</a> <font color="Maroon" title="c3">W</font>
<b>by </b><i><a class="txt" href="lmod_7.html#E2:58_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="lmod_7.html#T30" target="_self" title="LMOD_7:th.30">Th30</a></i>
<br/>.= 
<font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="lmod_7.html#E2:58_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:58"/>
<font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Maroon" title="c3">W</font> is  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a> <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">K</font>
 <b>by </b><i><a class="txt" href="lmod_7.html#E55"><i><font color="Green" title="E25">Lm2</font></i></a>, <a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>;<br/>


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