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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">v</font>, <font color="Maroon" title="c3">u</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/><b>let </b><font color="Maroon" title="c4">W1</font>, <font color="Maroon" title="c5">W2</font> be  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">strict</a>  <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>;<br/>







<b>assume </b><a NAME="E1:73"/><i><font color="Green" title="E54">A1</font></i>: 
<font color="Maroon" title="c2">v</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c4">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c5">W2</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c6">V1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W1</font>;<br/>
<b>set </b><font color="Maroon" title="c7">V2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">W2</font>;<br/>
<b>assume </b><a NAME="E2:73"/><i><font color="Green" title="E55">A2</font></i>: 
<font color="Maroon" title="c4">W1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">W2</font>
 ;<br/>

<a NAME="E3:73"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">W2</font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T38" target="_self" title="RLSUB_1:th.38">Th38</a></i>;<br/>
<a NAME="E4:73"/><b>then </b><i><font color="Green" title="E56">A3</font></i>: 
(  not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">W2</font> or  not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">W2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W1</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<div><i><font color="Green" title="E57">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:73_1"/><i><font color="Green" title="E58">A5</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">W2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon" title="c8">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">W2</font>;<br/><a NAME="E3:73_1"/>
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W1</font> &amp; not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">W2</font> )
 <b>by </b><i><a class="txt" href="rlsub_1.html#E1:73_1"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:73_1"/><b>then </b><i><font color="Green" title="E59">A6</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">W1</font> &amp; not <font color="Maroon" title="c8">x</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c5">W2</font> )
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/><a NAME="E5:73_1"/><b>then </b>
<font color="Maroon" title="c8">x</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T17" target="_self" title="RLSUB_1:th.17">Th17</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c8">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/><b>set </b><font color="Maroon" title="c10">z</font> = <font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">x</font>;<br/><a NAME="E7:73_1"/>
<font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b1">u2</font></span>)</span> where <font color="Olive" title="b1">u2</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">u2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">W1</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="rlsub_1.html#E4:73_1"><i><font color="Green" title="E59">A6</font></i></a></i>;<br/><a NAME="E8:73_1"/><b>then </b>
<font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">u</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c5">W2</font>
 <b>by </b><i><a class="txt" href="rlsub_1.html#E1:73"><i><font color="Green" title="E54">A1</font></i></a></i>;<br/><a NAME="E9:73_1"/><b>then </b>
<font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b1">u2</font></span>)</span> where <font color="Olive" title="b1">u2</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">u2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c5">W2</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c11">u1</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E11:73_1"/><i><font color="Green" title="E60">A7</font></i>: 
<font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c11">u1</font>
 <b>and </b><br/><a NAME="E12:73_1"/><i><font color="Green" title="E61">A8</font></i>: 
<font color="Maroon" title="c11">u1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c5">W2</font>
 ;<br/><font color="Maroon" title="c9">x</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">x</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">x</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c11">u1</font></span>)</span>
<b>by </b><i><a class="txt" href="rlsub_1.html#E11:73_1"><i><font color="Green" title="E60">A7</font></i></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
;<br/><a NAME="E14:73_1"/><b>then </b><i><font color="Green" title="E62">A9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c11">u1</font></span>)</span></span>)</span></span>)</span> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c4">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c4">W1</font>
 <b>by </b><i><a class="txt" href="rlsub_1.html#E4:73_1"><i><font color="Green" title="E59">A6</font></i></a>, <a class="ref" href="rlsub_1.html#T68" target="_self" title="RLSUB_1:th.68">Th68</a></i>;<br/><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c11">u1</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c11">u1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c11">u1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>
<br/>.= 
<font color="Maroon" title="c3">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c11">u1</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
;<br/><a NAME="E16:73_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c11">u1</font></span>)</span> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c5">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c11">u1</font></span>)</span> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c4">W1</font>
 <b>by </b><i><a class="txt" href="rlsub_1.html#E1:73"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="rlsub_1.html#E12:73_1"><i><font color="Green" title="E61">A8</font></i></a>, <a class="txt" href="rlsub_1.html#E14:73_1"><i><font color="Green" title="E62">A9</font></i></a>, <a class="ref" href="rlsub_1.html#T68" target="_self" title="RLSUB_1:th.68">Th68</a></i>;<br/><b>hence </b><a NAME="E17:73_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rlsub_1.html#E2:73"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="rlsub_1.html#T84" target="_self" title="RLSUB_1:th.84">Th84</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:73_2"/><i><font color="Green" title="E58">A10</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">W2</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon" title="c8">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">W2</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W1</font>;<br/><a NAME="E3:73_2"/>
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">W2</font> &amp; not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W1</font> )
 <b>by </b><i><a class="txt" href="rlsub_1.html#E1:73_2"><i><font color="Green" title="E58">A10</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:73_2"/><b>then </b><i><font color="Green" title="E59">A11</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c5">W2</font> &amp; not <font color="Maroon" title="c8">x</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">W1</font> )
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/><a NAME="E5:73_2"/><b>then </b>
<font color="Maroon" title="c8">x</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T17" target="_self" title="RLSUB_1:th.17">Th17</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c8">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/><b>set </b><font color="Maroon" title="c10">z</font> = <font color="Maroon" title="c3">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">x</font>;<br/><a NAME="E7:73_2"/>
<font color="Maroon" title="c3">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b1">u2</font></span>)</span> where <font color="Olive" title="b1">u2</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">u2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c5">W2</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="rlsub_1.html#E4:73_2"><i><font color="Green" title="E59">A11</font></i></a></i>;<br/><a NAME="E8:73_2"/><b>then </b>
<font color="Maroon" title="c3">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">v</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c4">W1</font>
 <b>by </b><i><a class="txt" href="rlsub_1.html#E1:73"><i><font color="Green" title="E54">A1</font></i></a></i>;<br/><a NAME="E9:73_2"/><b>then </b>
<font color="Maroon" title="c3">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b1">u2</font></span>)</span> where <font color="Olive" title="b1">u2</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">u2</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">W1</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c11">u1</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E11:73_2"/><i><font color="Green" title="E60">A12</font></i>: 
<font color="Maroon" title="c3">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c11">u1</font>
 <b>and </b><br/><a NAME="E12:73_2"/><i><font color="Green" title="E61">A13</font></i>: 
<font color="Maroon" title="c11">u1</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c4">W1</font>
 ;<br/><font color="Maroon" title="c9">x</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">x</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">x</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c11">u1</font></span>)</span>
<b>by </b><i><a class="txt" href="rlsub_1.html#E11:73_2"><i><font color="Green" title="E60">A12</font></i></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
;<br/><a NAME="E14:73_2"/><b>then </b><i><font color="Green" title="E62">A14</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c11">u1</font></span>)</span></span>)</span></span>)</span> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c5">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c5">W2</font>
 <b>by </b><i><a class="txt" href="rlsub_1.html#E4:73_2"><i><font color="Green" title="E59">A11</font></i></a>, <a class="ref" href="rlsub_1.html#T68" target="_self" title="RLSUB_1:th.68">Th68</a></i>;<br/><font color="Maroon" title="c3">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c11">u1</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c11">u1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c11">u1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>
<br/>.= 
<font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c11">u1</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
;<br/><a NAME="E16:73_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c11">u1</font></span>)</span> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c4">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c11">u1</font></span>)</span> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c5">W2</font>
 <b>by </b><i><a class="txt" href="rlsub_1.html#E1:73"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="rlsub_1.html#E12:73_2"><i><font color="Green" title="E61">A13</font></i></a>, <a class="txt" href="rlsub_1.html#E14:73_2"><i><font color="Green" title="E62">A14</font></i></a>, <a class="ref" href="rlsub_1.html#T68" target="_self" title="RLSUB_1:th.68">Th68</a></i>;<br/><b>hence </b><a NAME="E17:73_2"/>
contradiction
 <b>by </b><i><a class="txt" href="rlsub_1.html#E2:73"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="rlsub_1.html#T84" target="_self" title="RLSUB_1:th.84">Th84</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:73"/>
contradiction
 <b>by </b><i><a class="txt" href="rlsub_1.html#E4:73"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="rlsub_1.html#E5:73"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/>


</div>
