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<div class="add">

<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/>

<b>given </b><font color="Maroon">u</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">v</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><a NAME="E1:20"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a>  st <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> holds <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<a NAME="E2:20"/><i><font color="Green">E36</font></i>: 
( <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>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:20"/><i><font color="Green">E41</font></i>: 
not  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>,<font color="Maroon">u</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">v</font>, <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1"/><i><font color="Green">E43</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>,<font color="Maroon">u</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">v</font>, <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><a NAME="E1:20_1_1"/><i><font color="Green">E44</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span> )
 ;<br/><a NAME="E2:20_1_1"/>
( <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span>)</span> &amp; <font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span> )
 <b>by </b><i><a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a>, <a class="ref" href="rlvect_1.html#T27">RLVECT_1:27</a></i>;<br/><a NAME="E3:20_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
 <b>by </b><i><a class="txt" href="analoaf.html#E3"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="rlvect_1.html#T39">RLVECT_1:39</a></i>;<br/><a NAME="E4:20_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>;<br/><b>hence </b><a NAME="E5:20_1_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="analoaf.html#E3"><i><font color="Green">Lemma28</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:20_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="analoaf.html#E2"><i><font color="Green">Lemma27</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:20"/>
not  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>,<font color="Maroon">u</font> <a href="analoaf.html#R1" target="_self">//</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>,<font color="Maroon">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_2"/><i><font color="Green">E45</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>,<font color="Maroon">u</font> <a href="analoaf.html#R1" target="_self">//</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>,<font color="Maroon">v</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><a NAME="E1:20_2_1"/><i><font color="Green">E46</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span>)</span> )
 ;<br/><a NAME="E2:20_2_1"/>
( <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span>)</span> &amp; <font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>;<br/><b>then </b> <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i><a class="txt" href="analoaf.html#E3"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T39">RLVECT_1:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">b</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#T38">RLVECT_1:38</a></i>
;<br/><b>hence </b><a NAME="E4:20_2_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="analoaf.html#E3"><i><font color="Green">Lemma28</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:20_2"/>
contradiction
 <b>by </b><i>, <a class="txt" href="analoaf.html#E2"><i><font color="Green">Lemma27</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:20"/>
 ex <font color="Olive">u</font>, <font color="Olive">v</font>, <font color="Olive">w</font>, <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( not <font color="Olive">u</font>,<font color="Olive">v</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Olive">w</font>,<font color="Olive">y</font> &amp; not <font color="Olive">u</font>,<font color="Olive">v</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Olive">y</font>,<font color="Olive">w</font> )
 <b>by </b><i><a class="ref" href="analoaf.html#D1" target="_self">Def1</a></i>;<br/>


</div>
