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<b>let </b><font color="Maroon" title="c1">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c4">w</font>, <font color="Maroon" title="c5">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">V</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:31"/><i><font color="Green" title="E11">A1</font></i>: 
( <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> is <a href="rlvect_3.html#V1" title="RLVECT_3:attr.1">linearly-independent</a> &amp; <font color="Maroon" title="c3">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v</font> &amp; <font color="Maroon" title="c3">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">w</font> &amp; <font color="Maroon" title="c5">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">w</font> )
 <b>and </b><br/><a NAME="E2:31"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<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" title="c6">b</font>, <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c8">d</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:31_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c3">u</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="c7">c</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">d</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c5">v</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font>
 ;<br/><a NAME="E2:31_1"/><b>then </b>
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c3">u</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="c7">c</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">d</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c5">v</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>;<br/><a NAME="E3:31_1"/><b>then </b>
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c8">d</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="txt" href="rlvect_4.html#E1:31"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="rlvect_4.html#T10" target="_self" title="RLVECT_4:th.10">Th10</a></i>;<br/><b>hence </b><a NAME="E4:31_1"/>
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c8">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="txt" href="rlvect_4.html#E2:31"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:31"/>
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c4">w</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c5">v</font></span>)</span></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> is <a href="rlvect_3.html#V1" title="RLVECT_3:attr.1">linearly-independent</a>
 <b>by </b><i><a class="ref" href="rlvect_4.html#T10" target="_self" title="RLVECT_4:th.10">Th10</a></i>;<br/>


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