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<b>let </b><font color="Maroon" title="c6">S1</font>, <font color="Maroon" title="c7">S2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K13" title="LMOD_7:func.13">.</a> <font color="Maroon" title="c3">W</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">a</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font>  st <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c3">W</font> holds <br/><font color="Maroon" title="c6">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c3">W</font> ) &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font>  st <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c3">W</font> holds <br/><font color="Maroon" title="c7">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c3">W</font> ) implies <font color="Maroon" title="c6">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:50_1_2"/><i><font color="Green" title="E19">A3</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font>  st <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c3">W</font> holds <br/><font color="Maroon" title="c6">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c3">W</font>
 <b>and </b><br/><a NAME="E2:50_1_2"/><i><font color="Green" title="E20">A4</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font>  st <font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c3">W</font> holds <br/><font color="Maroon" title="c7">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c3">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S2</font></span><br/>

<b>consider </b><font color="Maroon" title="c8">a1</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E4:50_1_2"/><i><font color="Green" title="E21">A5</font></i>: 
<font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a1</font> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c3">W</font>
 <b>by </b><i><a class="ref" href="lmod_7.html#T22" target="_self" title="LMOD_7:th.22">Th22</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c6">S1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c8">a1</font></span>)</span> <a href="lmod_7.html#K14" title="LMOD_7:func.14">.</a> <font color="Maroon" title="c3">W</font>
<b>by </b><i><a class="txt" href="lmod_7.html#E1:50_1_2"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="lmod_7.html#E4:50_1_2"><i><font color="Green" title="E21">A5</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c7">S2</font>
<b>by </b><i><a class="txt" href="lmod_7.html#E2:50_1_2"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="lmod_7.html#E4:50_1_2"><i><font color="Green" title="E21">A5</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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