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<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">w</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:31"/><i><font color="Green">E45</font></i>: 
( <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">w</font>,<font color="Maroon">v</font></span><a href="domain_1.html#K8">}</a></span> is <a href="rlvect_3.html#V1">linearly-independent</a> &amp; <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">w</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">w</font> )
 <b>and </b><br/><a NAME="E2:31"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&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">b</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="real_1.html#NM1">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">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 ;<br/><a NAME="E2:31_1"/><b>then </b>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>;<br/><a NAME="E3:31_1"/><b>then </b>
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">d</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:31_1"/>
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T6">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">{</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">w</font>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span><a href="domain_1.html#K8">}</a></span> is <a href="rlvect_3.html#V1">linearly-independent</a>
 <b>by </b><i/>;<br/>


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