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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>;<br/>







<b>assume </b><a NAME="E1:19"/><i><font color="Green" title="E17">A1</font></i>: 
( <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> or <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> or <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> )
 ;<br/>

<a NAME="E2:19"/><i><font color="Green" title="E18">A2</font></i>: 
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> holds <br/><font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">w</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c6">v</font>, <font color="Maroon" title="c7">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>;<br/>





<b>assume </b><a NAME="E1:19_1"/><i><font color="Green" title="E19">A3</font></i>: 
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font>
 ;<br/>

 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span>
<b>by </b><i><a class="txt" href="anproj_1.html#E1:19_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="rlvect_1.html#T23" title="RLVECT_1:th.23">RLVECT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">v</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23" title="RLVECT_1:th.23">RLVECT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">w</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23" title="RLVECT_1:th.23">RLVECT_1:23</a></i>
;<br/>
<b>hence </b><a NAME="E3:19_1"/>
<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c7">w</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="ref" href="anproj_1.html#D3" target="_self" title="ANPROJ_1:def.3">Def3</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E19">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_2"/>
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font>
 ;<br/><a NAME="E2:19_2"/><b>then </b>
<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">u</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_1.html#E2:19"><i><font color="Green" title="E18">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E3:19_2"/>
<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="ref" href="anproj_1.html#T10" target="_self" title="ANPROJ_1:th.10">Th10</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_3"/>
<font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font>
 ;<br/><a NAME="E2:19_3"/><b>then </b>
<font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_1.html#E2:19"><i><font color="Green" title="E18">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E3:19_3"/>
<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="ref" href="anproj_1.html#T10" target="_self" title="ANPROJ_1:th.10">Th10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:19"/>
<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_1.html#E1:19"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="anproj_1.html#E2:19"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="anproj_1.html#E3:19"><i><font color="Green" title="E19">A4</font></i></a></i>;<br/>


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