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<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/>





<b>assume </b><a NAME="E1:59"/><i><font color="Green">E39</font></i>: 
 <a href="analmetr.html#R1">Gen</a> <font color="Maroon">x</font>,<font color="Maroon">y</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>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">u1</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">v1</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">w1</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/>









<b>consider </b><font color="Maroon">u</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:59"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">u</font>
 <b>and </b><br/><a NAME="E4:59"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">v</font>,<font color="Maroon">v1</font>,<font color="Maroon">v</font>,<font color="Maroon">u</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/>
<b>assume </b><a NAME="E5:59"/>
( not <font color="Maroon">v</font>,<font color="Maroon">v1</font>,<font color="Maroon">w</font>,<font color="Maroon">u1</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; not <font color="Maroon">v</font>,<font color="Maroon">v1</font>,<font color="Maroon">u1</font>,<font color="Maroon">w</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">u1</font>,<font color="Maroon">w1</font>,<font color="Maroon">u1</font>,<font color="Maroon">w</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 ;<br/>

<a NAME="E6:59"/><b>then </b><i><font color="Green">E45</font></i>: 
( not  <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v</font>, <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v1</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">w</font>,<font color="Maroon">u1</font> &amp;  <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v</font>, <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v1</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">v</font>,<font color="Maroon">u</font> &amp;  <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">u1</font>, <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">w1</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u1</font>,<font color="Maroon">w</font> &amp; not  <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v</font>, <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v1</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u1</font>,<font color="Maroon">w</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:59"/><b>then </b><i><font color="Green">E46</font></i>: 
( <font color="Maroon">v</font>,<font color="Maroon">u</font> <a href="analoaf.html#R1">//</a>  <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v</font>, <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v1</font> &amp; <font color="Maroon">u1</font>,<font color="Maroon">w</font> <a href="analoaf.html#R1">//</a>  <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">u1</font>, <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">w1</font> )
 
<b>by </b><i><a class="ref" href="analoaf.html#T21">ANALOAF:21</a></i>;<br/>
<a NAME="E8:59"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">u1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">w</font>
 
<b>by </b><i>, <a class="ref" href="analoaf.html#T18">ANALOAF:18</a></i>;<br/>
<a NAME="E9:59"/><i><font color="Green">E48</font></i>: 
( not <font color="Maroon">v</font>,<font color="Maroon">u</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u1</font>,<font color="Maroon">w</font> &amp; not <font color="Maroon">v</font>,<font color="Maroon">u</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">w</font>,<font color="Maroon">u1</font> )
 
<b>by </b><i>, , , <a class="ref" href="analoaf.html#T20">ANALOAF:20</a></i>;<br/>
<a NAME="E10:59"/>
(  <a href="analmetr.html#R1">Gen</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> implies  ex <font color="Olive">u</font>, <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>for <font color="Olive">w</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  ex <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="Olive">u</font></span>)</span> <a href="analort.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">w</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:59_1"/><i><font color="Green">E49</font></i>: 
 <a href="analmetr.html#R1">Gen</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 ;<br/>

<b>take </b>
<font color="Maroon">x</font>
;<br/>

<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>thus </b><a NAME="E2:59_1"/>
for <font color="Olive">w</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  ex <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">x</font></span>)</span> <a href="analort.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">w</font>
 <b>by </b><i><a class="txt" href="analort.html#E1:59"><i><font color="Green">E39</font></i></a>, <a class="ref" href="analmetr.html#D1">ANALMETR:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">u2</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E12:59"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">v</font>,<font color="Maroon">u</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">v</font>,<font color="Maroon">u2</font> or <font color="Maroon">v</font>,<font color="Maroon">u</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u2</font>,<font color="Maroon">v</font> )
 <b>and </b><br/><a NAME="E13:59"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">u1</font>,<font color="Maroon">w</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u1</font>,<font color="Maroon">u2</font> or <font color="Maroon">u1</font>,<font color="Maroon">w</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u2</font>,<font color="Maroon">u1</font> )
 <b>by </b><i>, <a class="txt" href="analort.html#E1"><i><font color="Green">Lemma38</font></i></a>, <a class="ref" href="analoaf.html#T31">ANALOAF:31</a></i>;<br/>
<a NAME="E14:59"/>
(  <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v</font>, <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v1</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">v</font>,<font color="Maroon">u2</font> or  <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v</font>, <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v1</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u2</font>,<font color="Maroon">v</font> )
 
<b>by </b><i>, , <a class="txt" href="analort.html#E1:59"><i><font color="Green">E39</font></i></a>, <a class="ref" href="analoaf.html#T20">ANALOAF:20</a></i>;<br/>
<a NAME="E15:59"/><b>then </b><i><font color="Green">E81</font></i>: 
( <font color="Maroon">v</font>,<font color="Maroon">v1</font>,<font color="Maroon">v</font>,<font color="Maroon">u2</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> or <font color="Maroon">v</font>,<font color="Maroon">v1</font>,<font color="Maroon">u2</font>,<font color="Maroon">v</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E16:59"/>
(  <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">u1</font>, <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">w1</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u1</font>,<font color="Maroon">u2</font> or  <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">u1</font>, <a href="analort.html#K3" target="_self">Orte</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">w1</font> <a href="analoaf.html#R1">//</a> <font color="Maroon">u2</font>,<font color="Maroon">u1</font> )
 
<b>by </b><i>, , <a class="txt" href="analort.html#E3:59"><i><font color="Green">E40</font></i></a>, <a class="ref" href="analoaf.html#T20">ANALOAF:20</a></i>;<br/>
<a NAME="E17:59"/><b>then </b>
( <font color="Maroon">u1</font>,<font color="Maroon">w1</font>,<font color="Maroon">u1</font>,<font color="Maroon">u2</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> or <font color="Maroon">u1</font>,<font color="Maroon">w1</font>,<font color="Maroon">u2</font>,<font color="Maroon">u1</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E18:59"/>
 ex <font color="Olive">u2</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( ( <font color="Maroon">v</font>,<font color="Maroon">v1</font>,<font color="Maroon">v</font>,<font color="Olive">u2</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> or <font color="Maroon">v</font>,<font color="Maroon">v1</font>,<font color="Olive">u2</font>,<font color="Maroon">v</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> ) &amp; ( <font color="Maroon">u1</font>,<font color="Maroon">w1</font>,<font color="Maroon">u1</font>,<font color="Olive">u2</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> or <font color="Maroon">u1</font>,<font color="Maroon">w1</font>,<font color="Olive">u2</font>,<font color="Maroon">u1</font> <a href="analort.html#R1" target="_self">are_COrte_wrt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> ) )
 <b>by </b><i><a class="txt" href="analort.html#E2"><i><font color="Green">Lemma41</font></i></a></i>;<br/>


</div>
