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

<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">x</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>assume </b><a NAME="E1:61"/><i><font color="Green" title="E32">A1</font></i>: 
 <a href="analmetr.html#R1" title="ANALMETR:pred.1">Gen</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c5">w</font>, <font color="Maroon" title="c6">u1</font>, <font color="Maroon" title="c7">v1</font>, <font color="Maroon" title="c8">w1</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>consider </b><font color="Maroon" title="c9">u</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="E3:61"/><i><font color="Green" title="E33">A2</font></i>: 
<font color="Maroon" title="c4">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">u</font>
 <b>and </b><br/><a NAME="E4:61"/><i><font color="Green" title="E34">A3</font></i>: 
<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c7">v1</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c9">u</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="txt" href="analort.html#E1:61"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="analort.html#T41" target="_self" title="ANALORT:th.41">Th41</a></i>;<br/>
<b>assume </b><a NAME="E5:61"/>
( not <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c7">v1</font>,<font color="Maroon" title="c5">w</font>,<font color="Maroon" title="c6">u1</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> &amp; not <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c7">v1</font>,<font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c5">w</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c8">w1</font>,<font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c5">w</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> )
 ;<br/>

<a NAME="E6:61"/><b>then </b><i><font color="Green" title="E35">A4</font></i>: 
( not  <a href="analort.html#K2" title="ANALORT:func.2">Ortm</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">v</font>, <a href="analort.html#K2" title="ANALORT:func.2">Ortm</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c7">v1</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c5">w</font>,<font color="Maroon" title="c6">u1</font> &amp;  <a href="analort.html#K2" title="ANALORT:func.2">Ortm</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">v</font>, <a href="analort.html#K2" title="ANALORT:func.2">Ortm</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c7">v1</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c9">u</font> &amp;  <a href="analort.html#K2" title="ANALORT:func.2">Ortm</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c6">u1</font>, <a href="analort.html#K2" title="ANALORT:func.2">Ortm</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c8">w1</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c5">w</font> &amp; not  <a href="analort.html#K2" title="ANALORT:func.2">Ortm</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">v</font>, <a href="analort.html#K2" title="ANALORT:func.2">Ortm</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c7">v1</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c5">w</font> )
 
<b>by </b><i><a class="txt" href="analort.html#E4:61"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="analort.html#D4" target="_self" title="ANALORT:def.4">Def4</a></i>;<br/>
<a NAME="E7:61"/><b>then </b><i><font color="Green" title="E36">A5</font></i>: 
( <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c9">u</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a>  <a href="analort.html#K2" title="ANALORT:func.2">Ortm</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">v</font>, <a href="analort.html#K2" title="ANALORT:func.2">Ortm</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c7">v1</font> &amp; <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c5">w</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a>  <a href="analort.html#K2" title="ANALORT:func.2">Ortm</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c6">u1</font>, <a href="analort.html#K2" title="ANALORT:func.2">Ortm</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c8">w1</font> )
 
<b>by </b><i><a class="ref" href="analoaf.html#T21" title="ANALOAF:th.21">ANALOAF:21</a></i>;<br/>
<a NAME="E8:61"/><i><font color="Green" title="E37">A6</font></i>: 
<font color="Maroon" title="c6">u1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">w</font>
 
<b>by </b><i><a class="txt" href="analort.html#E6:61"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="analoaf.html#T18" title="ANALOAF:th.18">ANALOAF:18</a></i>;<br/>
<a NAME="E9:61"/><i><font color="Green" title="E38">A7</font></i>: 
( not <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c9">u</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c5">w</font> &amp; not <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c9">u</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c5">w</font>,<font color="Maroon" title="c6">u1</font> )
 
<b>by </b><i><a class="txt" href="analort.html#E3:61"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="analort.html#E6:61"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="analort.html#E7:61"><i><font color="Green" title="E36">A5</font></i></a>, <a class="ref" href="analoaf.html#T20" title="ANALOAF:th.20">ANALOAF:20</a></i>;<br/>
<a NAME="E10:61"/>
(  <a href="analmetr.html#R1" title="ANALMETR:pred.1">Gen</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> implies  ex <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> st <br/>for <font color="Olive" title="b3">w</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  ex <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <span class="p1">(<span class="default"><font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="analort.html#K1" title="ANALORT:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">b</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Olive" title="b2">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">w</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_1"/><i><font color="Green" title="E39">A8</font></i>: 
 <a href="analmetr.html#R1" title="ANALMETR:pred.1">Gen</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>
 ;<br/>

<b>take </b>
<font color="Maroon" title="c2">x</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c3">y</font>
;<br/>

<b>thus </b><a NAME="E2:61_1"/>
for <font color="Olive" title="b1">w</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  ex <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="analort.html#K1" title="ANALORT:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">b</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">w</font>
 <b>by </b><i><a class="txt" href="analort.html#E1:61_1"><i><font color="Green" title="E39">A8</font></i></a>, <a class="ref" href="analmetr.html#D1" title="ANALMETR:def.1">ANALMETR:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c10">u2</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="E12:61"/><i><font color="Green" title="E39">A9</font></i>: 
( <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c9">u</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c10">u2</font> or <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c9">u</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c10">u2</font>,<font color="Maroon" title="c4">v</font> )
 <b>and </b><br/><a NAME="E13:61"/><i><font color="Green" title="E40">A10</font></i>: 
( <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c5">w</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c10">u2</font> or <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c5">w</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c10">u2</font>,<font color="Maroon" title="c6">u1</font> )
 <b>by </b><i><a class="txt" href="analort.html#E1:61"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="analort.html#E9:61"><i><font color="Green" title="E38">A7</font></i></a>, <a class="ref" href="analoaf.html#T31" title="ANALOAF:th.31">ANALOAF:31</a></i>;<br/>
<a NAME="E14:61"/>
(  <a href="analort.html#K2" title="ANALORT:func.2">Ortm</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">v</font>, <a href="analort.html#K2" title="ANALORT:func.2">Ortm</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c7">v1</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c10">u2</font> or  <a href="analort.html#K2" title="ANALORT:func.2">Ortm</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">v</font>, <a href="analort.html#K2" title="ANALORT:func.2">Ortm</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c7">v1</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c10">u2</font>,<font color="Maroon" title="c4">v</font> )
 
<b>by </b><i><a class="txt" href="analort.html#E3:61"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="analort.html#E7:61"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="analort.html#E12:61"><i><font color="Green" title="E39">A9</font></i></a>, <a class="ref" href="analoaf.html#T20" title="ANALOAF:th.20">ANALOAF:20</a></i>;<br/>
<a NAME="E15:61"/><b>then </b><i><font color="Green" title="E41">A11</font></i>: 
( <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c7">v1</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c10">u2</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> or <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c7">v1</font>,<font color="Maroon" title="c10">u2</font>,<font color="Maroon" title="c4">v</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> )
 
<b>by </b><i><a class="ref" href="analort.html#D4" target="_self" title="ANALORT:def.4">Def4</a></i>;<br/>
<a NAME="E16:61"/>
(  <a href="analort.html#K2" title="ANALORT:func.2">Ortm</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c6">u1</font>, <a href="analort.html#K2" title="ANALORT:func.2">Ortm</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c8">w1</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c10">u2</font> or  <a href="analort.html#K2" title="ANALORT:func.2">Ortm</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c6">u1</font>, <a href="analort.html#K2" title="ANALORT:func.2">Ortm</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c8">w1</font> <a href="analoaf.html#R1" title="ANALOAF:pred.1">//</a> <font color="Maroon" title="c10">u2</font>,<font color="Maroon" title="c6">u1</font> )
 
<b>by </b><i><a class="txt" href="analort.html#E7:61"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="analort.html#E8:61"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="analort.html#E13:61"><i><font color="Green" title="E40">A10</font></i></a>, <a class="ref" href="analoaf.html#T20" title="ANALOAF:th.20">ANALOAF:20</a></i>;<br/>
<a NAME="E17:61"/><b>then </b>
( <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c8">w1</font>,<font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c10">u2</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> or <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c8">w1</font>,<font color="Maroon" title="c10">u2</font>,<font color="Maroon" title="c6">u1</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> )
 
<b>by </b><i><a class="ref" href="analort.html#D4" target="_self" title="ANALORT:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E18:61"/>
 ex <font color="Olive" title="b1">u2</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> st <br/>( ( <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c7">v1</font>,<font color="Maroon" title="c4">v</font>,<font color="Olive" title="b1">u2</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> or <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c7">v1</font>,<font color="Olive" title="b1">u2</font>,<font color="Maroon" title="c4">v</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> ) &amp; ( <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c8">w1</font>,<font color="Maroon" title="c6">u1</font>,<font color="Olive" title="b1">u2</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> or <font color="Maroon" title="c6">u1</font>,<font color="Maroon" title="c8">w1</font>,<font color="Olive" title="b1">u2</font>,<font color="Maroon" title="c6">u1</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> ) )
 <b>by </b><i><a class="txt" href="analort.html#E15:61"><i><font color="Green" title="E41">A11</font></i></a></i>;<br/>


</div>
