<?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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b3">W</font> being   <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> holds <br/> (  ex <font color="Olive" title="b4">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/>( <font color="Olive" title="b1">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">v</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b2">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">v</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Olive" title="b3">W</font> ) iff <font color="Olive" title="b1">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b2">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b3">W</font> )</span><br/><b>let </b><font color="Maroon" title="c2">v1</font>, <font color="Maroon" title="c3">v2</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">W</font> being   <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> holds <br/> (  ex <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/>( <font color="Maroon" title="c2">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">v</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Olive" title="b1">W</font> &amp; <font color="Maroon" title="c3">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">v</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Olive" title="b1">W</font> ) iff <font color="Maroon" title="c2">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">W</font> )</span><br/><b>let </b><font color="Maroon" title="c4">W</font> be    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">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/>( <font color="Maroon" title="c2">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">v</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c4">W</font> &amp; <font color="Maroon" title="c3">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">v</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c4">W</font> ) iff <font color="Maroon" title="c2">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W</font> )</span><br/>







<b>thus </b><a NAME="E1:68"/>
(  ex <font color="Olive" title="b1">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/>( <font color="Maroon" title="c2">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">v</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c4">W</font> &amp; <font color="Maroon" title="c3">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">v</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c4">W</font> ) implies <font color="Maroon" title="c2">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W</font> implies  ex <font color="Olive" title="b1">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/>( <font color="Maroon" title="c2">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">v</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c4">W</font> &amp; <font color="Maroon" title="c3">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">v</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c4">W</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c5">v</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><a NAME="E2:68_1"/><i><font color="Green" title="E49">A1</font></i>: 
<font color="Maroon" title="c2">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">v</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c4">W</font>
 <b>and </b><br/><a NAME="E3:68_1"/><i><font color="Green" title="E50">A2</font></i>: 
<font color="Maroon" title="c3">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">v</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c4">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W</font></span><br/>

<b>consider </b><font color="Maroon" title="c6">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="E5:68_1"/><i><font color="Green" title="E51">A3</font></i>: 
<font color="Maroon" title="c6">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W</font>
 <b>and </b><br/><a NAME="E6:68_1"/><i><font color="Green" title="E52">A4</font></i>: 
<font color="Maroon" title="c2">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">u1</font>
 <b>by </b><i><a class="txt" href="#E2:68_1"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="rlsub_1.html#T79" target="_self" title="RLSUB_1:th.79">Th79</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">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="E8:68_1"/><i><font color="Green" title="E53">A5</font></i>: 
<font color="Maroon" title="c7">u2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W</font>
 <b>and </b><br/><a NAME="E9:68_1"/><i><font color="Green" title="E54">A6</font></i>: 
<font color="Maroon" title="c3">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">u2</font>
 <b>by </b><i><a class="txt" href="#E3:68_1"><i><font color="Green" title="E50">A2</font></i></a>, <a class="ref" href="rlsub_1.html#T79" target="_self" title="RLSUB_1:th.79">Th79</a></i>;<br/>
<font color="Maroon" title="c2">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">v2</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c7">u2</font></span>)</span>
<b>by </b><i><a class="txt" href="#E6:68_1"><i><font color="Green" title="E52">A4</font></i></a>, <a class="txt" href="#E9:68_1"><i><font color="Green" title="E54">A6</font></i></a>, <a class="ref" href="rlvect_1.html#T44" title="RLVECT_1:th.44">RLVECT_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c7">u2</font>
<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"><font color="Maroon" title="c6">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">v</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c7">u2</font>
<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"><font color="Maroon" title="c6">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c7">u2</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></i>
<br/>.= 
<font color="Maroon" title="c6">u1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c7">u2</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
;<br/>
<b>hence </b><a NAME="E11:68_1"/>
<font color="Maroon" title="c2">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W</font>
 <b>by </b><i><a class="txt" href="#E5:68_1"><i><font color="Green" title="E51">A3</font></i></a>, <a class="txt" href="#E8:68_1"><i><font color="Green" title="E53">A5</font></i></a>, <a class="ref" href="rlsub_1.html#T31" target="_self" title="RLSUB_1:th.31">Th31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:68"/>
<font color="Maroon" title="c2">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">v2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">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/>( <font color="Maroon" title="c2">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">v</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c4">W</font> &amp; <font color="Maroon" title="c3">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">v</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c4">W</font> )</span><br/>

<a NAME="E3:68"/><b>then </b><i><font color="Green" title="E49">A7</font></i>: 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">v2</font></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W</font>
 
<b>by </b><i><a class="ref" href="rlsub_1.html#T30" target="_self" title="RLSUB_1:th.30">Th30</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c2">v1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">v1</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c4">W</font> &amp; <font color="Maroon" title="c3">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">v1</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c4">W</font> )</span><br/>

<b>thus </b><a NAME="E4:68"/>
<font color="Maroon" title="c2">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">v1</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c4">W</font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T59" target="_self" title="RLSUB_1:th.59">Th59</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">v1</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c4">W</font></span><br/>

<font color="Maroon" title="c2">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">v1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">v2</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c2">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">v2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T47" title="RLVECT_1:th.47">RLVECT_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">v2</font>
<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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">v2</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></i>
<br/>.= 
<font color="Maroon" title="c3">v2</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
;<br/>
<b>hence </b><a NAME="E6:68"/>
<font color="Maroon" title="c3">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">v1</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c4">W</font>
 <b>by </b><i><a class="txt" href="#E3:68"><i><font color="Green" title="E49">A7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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