<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">V</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">W1</font> be  <a href="bhsp_1.html#V1">strict</a>  <a href="rusub_1.html#M1" target="_self">Subspace</a> of <font color="Maroon">V</font>, <font color="Maroon">W2</font> be  <a href="bhsp_1.html#V1">strict</a>  <a href="rusub_1.html#M1" target="_self">Subspace</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/>





<b>assume </b><a NAME="E1:66"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">v</font> <a href="rusub_1.html#K3" target="_self">+</a> <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="rusub_1.html#K3" target="_self">+</a> <font color="Maroon">W2</font>
 ;<br/>

<b>set </b><font color="Maroon">V1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font>;<br/>
<b>set </b><font color="Maroon">V2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>;<br/>
<b>assume </b><a NAME="E2:66"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">W1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">W2</font>
 ;<br/>

<a NAME="E3:66"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>
 
<b>by </b><i><a class="txt" href="rusub_1.html#E2:66"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E4:66"/><b>then </b><i><font color="Green">E42</font></i>: 
(  not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> or  not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<div><i><font color="Green">E62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_1"/><i><font color="Green">E63</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="subset_1.html#K6">\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="subset_1.html#K6">\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font>;<br/><a NAME="E3:66_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> )
 <b>by </b><i><a class="ref" href="rusub_1.html#T16" target="_self">Th16</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:66_1"/><b>then </b><i><font color="Green">E64</font></i>: 
( <font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; not <font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> )
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E5:66_1"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><b>set </b><font color="Maroon">z</font> = <font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font>;<br/><a NAME="E7:66_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u2</font></span>)</span> where <font color="Olive">u2</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">u2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="rusub_1.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E8:66_1"/><b>then </b>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> <a href="rusub_1.html#K3" target="_self">+</a> <font color="Maroon">W2</font>
 <b>by </b><i/>;<br/><a NAME="E9:66_1"/><b>then </b>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u2</font></span>)</span> where <font color="Olive">u2</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">u2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">u1</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E11:66_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font>
 <b>and </b><br/><a NAME="E12:66_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">u1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font>
 ;<br/><font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font>
<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">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font></span>)</span>
<b>by </b><i><a class="ref" href="rusub_1.html#T18" target="_self">Th18</a>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
;<br/><a NAME="E14:66_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font></span>)</span></span>)</span></span>)</span> <a href="rusub_1.html#K3" target="_self">+</a> <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rusub_1.html#K3" target="_self">+</a> <font color="Maroon">W1</font>
 <b>by </b><i><a class="ref" href="rusub_1.html#T17" target="_self">Th17</a>, </i>;<br/><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/><a NAME="E16:66_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font></span>)</span> <a href="rusub_1.html#K3" target="_self">+</a> <font color="Maroon">W2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font></span>)</span> <a href="rusub_1.html#K3" target="_self">+</a> <font color="Maroon">W1</font>
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E17:66_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2"/><i><font color="Green">E72</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> <a href="subset_1.html#K6">\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> <a href="subset_1.html#K6">\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font>;<br/><a NAME="E3:66_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W2</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> )
 <b>by </b><i><a class="ref" href="rusub_1.html#T16" target="_self">Th16</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:66_2"/><b>then </b><i><font color="Green">E75</font></i>: 
( <font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> &amp; not <font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> )
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E5:66_2"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><b>set </b><font color="Maroon">z</font> = <font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font>;<br/><a NAME="E7:66_2"/>
<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u2</font></span>)</span> where <font color="Olive">u2</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">u2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="rusub_1.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E8:66_2"/><b>then </b>
<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="rusub_1.html#K3" target="_self">+</a> <font color="Maroon">W1</font>
 <b>by </b><i/>;<br/><a NAME="E9:66_2"/><b>then </b>
<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u2</font></span>)</span> where <font color="Olive">u2</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">u2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">u1</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E11:66_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font>
 <b>and </b><br/><a NAME="E12:66_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">u1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font>
 ;<br/><font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font>
<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">u</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font></span>)</span>
<b>by </b><i><a class="ref" href="rusub_1.html#T18" target="_self">Th18</a>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
;<br/><a NAME="E14:66_2"/><b>then </b><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font></span>)</span></span>)</span></span>)</span> <a href="rusub_1.html#K3" target="_self">+</a> <font color="Maroon">W2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="rusub_1.html#K3" target="_self">+</a> <font color="Maroon">W2</font>
 <b>by </b><i><a class="ref" href="rusub_1.html#T17" target="_self">Th17</a>, </i>;<br/><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/><a NAME="E16:66_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font></span>)</span> <a href="rusub_1.html#K3" target="_self">+</a> <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u1</font></span>)</span> <a href="rusub_1.html#K3" target="_self">+</a> <font color="Maroon">W2</font>
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E17:66_2"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:66"/>
contradiction
 <b>by </b><i><a class="ref" href="rusub_1.html#T14" target="_self">Th14</a>, <a class="ref" href="rusub_1.html#T15" target="_self">Th15</a>, <a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>


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