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

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>  st (  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font> or  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Olive" title="b3">w</font>,<font color="Olive" title="b1">u</font> or  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">w</font> ) holds <br/><font color="Olive" title="b3">w</font>,<font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> or  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">u</font> or  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> ) implies <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )</span><br/>







<a NAME="E1:19"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
 for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>  st  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font> holds <br/><font color="Olive" title="b3">w</font>,<font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c6">v</font>, <font color="Maroon" title="c7">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">v</font> implies <font color="Maroon" title="c7">w</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">v</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )</span><br/>





<a NAME="E1:19_1"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">w</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:19_1"/>
 <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">w</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">v</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:19_1"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v</font>
 <span class="kw">and </span><br/><a NAME="E5:19_1"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
<font color="Maroon" title="c8">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> 
 <span class="kw">and </span><br/><a NAME="E6:19_1"/>
<font color="Maroon" title="c9">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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="anproj_1.html#D1" target="_self" title="ANPROJ_1:def.1">Def2</a></span>;<br/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">u</font></span>)</span> <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="c9">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="rlvect_1.html#T5" title="RLVECT_1:th.5">RLVECT_1:5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">u</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="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">u</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"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></span>
;<br/>
<a NAME="E8:19_1"/><span class="kw">then </span>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">w</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">u</font></span>)</span></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"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="rlvect_1.html#T4" title="RLVECT_1:th.4">RLVECT_1:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:19_1"/>
<font color="Maroon" title="c7">w</font>,<font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">v</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="anproj_1.html#D2" target="_self" title="ANPROJ_1:def.2">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E19">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:19_2"/>
 <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">u</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> or  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">u</font> or  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> ) implies <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )</span><br/><a NAME="E2:19_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">u</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E18">A1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:19_2"/>
( (  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> or  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">u</font> or  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> ) implies <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="anproj_1.html#T5" target="_self" title="ANPROJ_1:th.5">Th10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E20">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:19_3"/>
 <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> or  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">u</font> or  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> ) implies <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )</span><br/><a NAME="E2:19_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E18">A1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:19_3"/>
( (  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> or  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">u</font> or  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> ) implies <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="anproj_1.html#T5" target="_self" title="ANPROJ_1:th.5">Th10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E4:19"/>
(  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> or  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">u</font> or  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> </span><br/>

<span class="kw">hence </span><a NAME="E5:19"/>
<font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E20">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
