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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">w</font>, <font color="Maroon" title="c3">y</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">p</font>, <font color="Maroon" title="c5">p1</font>, <font color="Maroon" title="c6">q</font>, <font color="Maroon" title="c7">q1</font>, <font color="Maroon" title="c8">r</font>, <font color="Maroon" title="c9">r1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" title="ANALMETR:func.2">AMSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font></span>)</span>;<br/>

















<b>assume </b><b>that </b><br/><a NAME="E1:41"/><i><font color="Green" title="E34">A1</font></i>: 
<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">p1</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font>
 <b>and </b><br/><a NAME="E2:41"/><i><font color="Green" title="E35">A2</font></i>: 
<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">p1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c9">r1</font>
 ;<br/>

<b>assume </b><a NAME="E3:41"/><i><font color="Green" title="E36">A3</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">p1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c10">u</font> = <font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c11">v</font> = <font color="Maroon" title="c5">p1</font>, <font color="Maroon" title="c12">u1</font> = <font color="Maroon" title="c6">q</font>, <font color="Maroon" title="c13">v1</font> = <font color="Maroon" title="c7">q1</font>, <font color="Maroon" title="c14">u2</font> = <font color="Maroon" title="c8">r</font>, <font color="Maroon" title="c15">v2</font> = <font color="Maroon" title="c9">r1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<a NAME="E5:41"/>
<font color="Maroon" title="c10">u</font>,<font color="Maroon" title="c11">v</font>,<font color="Maroon" title="c12">u1</font>,<font color="Maroon" title="c13">v1</font> <a href="analmetr.html#R3" title="ANALMETR:pred.3">are_Ort_wrt</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>
 
<b>by </b><i><a class="txt" href="analmetr.html#E1:41"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="analmetr.html#T31" target="_self" title="ANALMETR:th.31">Th31</a></i>;<br/>
<a NAME="E6:41"/><b>then </b><i><font color="Green" title="E37">A4</font></i>: 
<font color="Maroon" title="c11">v</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c10">u</font>,<font color="Maroon" title="c13">v1</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c12">u1</font> <a href="analmetr.html#R2" title="ANALMETR:pred.2">are_Ort_wrt</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D3" target="_self" title="ANALMETR:def.3">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">a</font>, <font color="Maroon" title="c17">b</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E8:41"/><i><font color="Green" title="E38">A5</font></i>: 
<font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">v</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c10">u</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">b</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">v2</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c14">u2</font></span>)</span>
 <b>and </b><br/><a NAME="E9:41"/><i><font color="Green" title="E39">A6</font></i>: 
( <font color="Maroon" title="c16">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c17">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="analmetr.html#E2:41"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="analmetr.html#T32" target="_self" title="ANALMETR:th.32">Th32</a></i>;<br/>
<a NAME="E10:41"/>
<font color="Maroon" title="c17">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1"/>
<font color="Maroon" title="c17">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<a NAME="E2:41_1"/><b>then </b>
( <font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">v</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c10">u</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> &amp; <font color="Maroon" title="c16">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="analmetr.html#E8:41"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="analmetr.html#E9:41"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="rlvect_1.html#T23" title="RLVECT_1:th.23">RLVECT_1:23</a></i>;<br/>
<a NAME="E3:41_1"/><b>then </b>
<font color="Maroon" title="c11">v</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T24" title="RLVECT_1:th.24">RLVECT_1:24</a></i>;<br/>
<b>hence </b><a NAME="E4:41_1"/>
contradiction
 <b>by </b><i><a class="txt" href="analmetr.html#E3:41"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="rlvect_1.html#T35" title="RLVECT_1:th.35">RLVECT_1:35</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><font color="Maroon" title="c15">v2</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c14">u2</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">v</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c10">u</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="analmetr.html#E8:41"><i><font color="Green" title="E38">A5</font></i></a>, <a class="ref" href="analoaf.html#T12" title="ANALOAF:th.12">ANALOAF:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">v</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c10">u</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
;<br/>
<a NAME="E12:41"/><b>then </b>
<font color="Maroon" title="c15">v2</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c14">u2</font>,<font color="Maroon" title="c13">v1</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c12">u1</font> <a href="analmetr.html#R2" title="ANALMETR:pred.2">are_Ort_wrt</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>
 
<b>by </b><i><a class="txt" href="analmetr.html#E6:41"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="analmetr.html#T11" target="_self" title="ANALMETR:th.11">Th11</a></i>;<br/>
<a NAME="E13:41"/><b>then </b>
<font color="Maroon" title="c13">v1</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c12">u1</font>,<font color="Maroon" title="c15">v2</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c14">u2</font> <a href="analmetr.html#R2" title="ANALMETR:pred.2">are_Ort_wrt</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T8" target="_self" title="ANALMETR:th.8">Th8</a></i>;<br/>
<a NAME="E14:41"/><b>then </b>
<font color="Maroon" title="c12">u1</font>,<font color="Maroon" title="c13">v1</font>,<font color="Maroon" title="c14">u2</font>,<font color="Maroon" title="c15">v2</font> <a href="analmetr.html#R3" title="ANALMETR:pred.3">are_Ort_wrt</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D3" target="_self" title="ANALMETR:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E15:41"/>
<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">q1</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c9">r1</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T31" target="_self" title="ANALMETR:th.31">Th31</a></i>;<br/>


</div>
