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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">w</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>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">u1</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">u2</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="E1:16"/><i><font color="Green">E38</font></i>: 
 <a href="analmetr.html#R1" target="_self">Gen</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E2:16"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">v</font>,<font color="Maroon">u1</font> <a href="analmetr.html#R2" target="_self">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E3:16"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">v</font>,<font color="Maroon">u2</font> <a href="analmetr.html#R2" target="_self">are_Ort_wrt</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E4:16"/><i><font color="Green">E51</font></i>: 
<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>
 ;<br/>

<b>consider </b><font color="Maroon">a1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">a2</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:16"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span> &amp; <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span> )
 <b>and </b><br/><a NAME="E7:16"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">a1'</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">a2'</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">c1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">c2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E9:16"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a1'</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a2'</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span> &amp; <font color="Maroon">u2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span> )
 <b>and </b><br/><a NAME="E10:16"/><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a1'</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">a2'</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c2</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, </i>;<br/>
<a NAME="E11:16"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1'</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2'</font> )
 
<b>by </b><i>, , , </i>;<br/>
<div><i><font color="Green">E65</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:16_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, , </i>;<br/><a NAME="E3:16_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">c2</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/><a NAME="E4:16_1"/>
<font color="Maroon">b2</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/><a NAME="E5:16_1"/><b>then </b><i><font color="Green">E69</font></i>: 
( <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> &amp; <font color="Maroon">u2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>;<br/><a NAME="E6:16_1"/><b>then </b><i><font color="Green">E70</font></i>: 
( <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font> &amp; <font color="Maroon">u2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font> )
 <b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/><i><font color="Green">E71</font></i>: <font color="Maroon">c1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font> = 
<font color="Maroon">c1</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<font color="Maroon">b1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1_2"/>
<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>then </b>1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font> = 
0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
;<br/><b>hence </b><a NAME="E3:16_1_2"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font> &amp; ( <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) )
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:16_1"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font> &amp; ( <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:16_2"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> &amp; <font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c2</font></span>)</span> )
 <b>by </b><i>, , </i>;<br/><i><font color="Green">E74</font></i>: <font color="Maroon">b1</font> = 
1 <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font>
<b>by </b><i>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K2">"</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a2</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K2">"</a> </span>)</span>
<b>by </b><i/>
;<br/><i><font color="Green">E75</font></i>: <font color="Maroon">c1</font> = 
1 <a href="real_1.html#K4">*</a> <font color="Maroon">c1</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c1</font>
<b>by </b><i>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K2">"</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a2</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K2">"</a> </span>)</span>
<b>by </b><i/>
;<br/><a NAME="E5:16_2"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">b2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b2</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a2</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E6:16_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c2</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a2</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a2</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c2</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span>
 ;<br/><div><i><font color="Green">E78</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_2_3"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">c2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">b2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><b>take </b><font color="Maroon">a</font> = <font color="Maroon">c2</font>;<br/><b>take </b><font color="Maroon">b</font> = <font color="Maroon">b2</font>;<br/><b>thus </b><a NAME="E2:16_2_3"/>
( <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font> &amp; ( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) )
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_2_4"/>
( <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c2</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><a NAME="E2:16_2_4"/><b>then </b>
1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> 1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font>
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E3:16_2_4"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font> &amp; ( <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) )
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:16_2"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font> &amp; ( <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:16"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font> &amp; ( <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) )
 <b>by </b><i/>;<br/>


</div>
