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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>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lmod_7.html#K17" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E2:58_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="lmod_7.html#K18" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="lmod_7.html#T28" target="_self">Th28</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:58_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="lmod_7.html#K18" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="lmod_7.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E5:58_1"/><i><font color="Green">E29</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="lmod_7.html#K18" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lmod_7.html#K18" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="lmod_7.html#K18" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lmod_7.html#K18" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="lmod_7.html#E2:58_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="lmod_7.html#E4:58_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="lmod_7.html#T30" target="_self">Th30</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> = 
<font color="Maroon">c<sub>4</sub></font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lmod_7.html#K18" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="lmod_7.html#E2:58_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="lmod_7.html#E4:58_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="lmod_7.html#T30" target="_self">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="vectsp_1.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="lmod_7.html#K18" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="lmod_7.html#T30" target="_self">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="lmod_7.html#K18" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="lmod_7.html#E5:58_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="lmod_7.html#T30" target="_self">Th30</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="lmod_7.html#K18" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="lmod_7.html#E2:58_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="lmod_7.html#T30" target="_self">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="lmod_7.html#K18" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="lmod_7.html#E5:58_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="lmod_7.html#T30" target="_self">Th30</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="lmod_7.html#K18" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="lmod_7.html#E2:58_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="lmod_7.html#T30" target="_self">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="vectsp_1.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="lmod_7.html#K18" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="lmod_7.html#K18" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lmod_7.html#T30" target="_self">Th30</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="lmod_7.html#E2:58_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="lmod_7.html#T30" target="_self">Th30</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="lmod_7.html#K18" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="lmod_7.html#E2:58_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="lmod_7.html#T30" target="_self">Th30</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="lmod_7.html#E2:58_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:58"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lmod_7.html#K17" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font> is  <a href="vectsp_1.html#V11">strict</a> <a href="vectsp_2.html#NM6">LeftMod</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="lmod_7.html#E55"><i><font color="Green">Lemma25</font></i></a>, <a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>;<br/>


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