<?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>, <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>  st  not <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">w</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font>,<font color="Olive" title="b4">p</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">w</font>,<font color="Olive" title="b5">q</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  holds <br/> ex <font color="Olive" title="b6">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b1">u</font>,<font color="Olive" title="b3">w</font>,<font color="Olive" title="b6">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Olive" title="b4">p</font>,<font color="Olive" title="b5">q</font>,<font color="Olive" title="b6">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp;  not <font color="Olive" title="b6">y</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</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>, <font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">q</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"> (  not <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>  &amp; <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c5">p</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c6">q</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  implies  ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp;  not <font color="Olive" title="b1">y</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  ) )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:24"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
 not <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">and </span><br/><a NAME="E2:24"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c5">p</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <span class="kw">and </span><br/><a NAME="E3:24"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c6">q</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp;  not <font color="Olive" title="b1">y</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  )</span><br/>

<a NAME="E4:24"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
 not <font color="Maroon" title="c3">v</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="anproj_1.html#T12" target="_self" title="ANPROJ_1:th.12">Th17</a></span>;<br/>
<a NAME="E5:24"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
 not <font color="Maroon" title="c4">w</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="anproj_1.html#T12" target="_self" title="ANPROJ_1:th.12">Th17</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E27">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><span class="kw">now </span></a><div class="add"><div><span class="lab"><font color="Green" title="E28">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:24_1_1"/>
<font color="Maroon" title="c6">q</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp;  not <font color="Olive" title="b1">y</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  )</span><br/><a NAME="E2:24_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">q</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="struct_0.html#D12" title="STRUCT_0:def.12">STRUCT_0:def 12</a></span>;<br/><a NAME="E3:24_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A8</font></span>: 
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</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="ref" href="anproj_1.html#T10" target="_self" title="ANPROJ_1:th.10">Th15</a></span>;<br/><a NAME="E4:24_1_1"/>
<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</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="ref" href="anproj_1.html#T11" target="_self" title="ANPROJ_1:th.11">Th16</a></span>;<br/><span class="kw">hence </span><a NAME="E5:24_1_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp;  not <font color="Olive" title="b1">y</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E3:24_1_1"><span class="lab"><font color="Green" title="E29">A8</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><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E29">A9</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:24_1_2"/>
<font color="Maroon" title="c5">p</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp;  not <font color="Olive" title="b1">y</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  )</span><br/><a NAME="E2:24_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</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="struct_0.html#D12" title="STRUCT_0:def.12">STRUCT_0:def 12</a></span>;<br/><a NAME="E3:24_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A10</font></span>: 
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</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="ref" href="anproj_1.html#T10" target="_self" title="ANPROJ_1:th.10">Th15</a></span>;<br/><a NAME="E4:24_1_2"/>
<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</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="ref" href="anproj_1.html#T11" target="_self" title="ANPROJ_1:th.11">Th16</a></span>;<br/><span class="kw">hence </span><a NAME="E5:24_1_2"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp;  not <font color="Olive" title="b1">y</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E3:24_1_2"><span class="lab"><font color="Green" title="E30">A10</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><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E30">A11</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:24_1_3"/>
 <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp;  not <font color="Olive" title="b1">y</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  )</span><br/><a NAME="E2:24_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A12</font></span>: 
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</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="ref" href="anproj_1.html#T11" target="_self" title="ANPROJ_1:th.11">Th16</a></span>;<br/><a NAME="E3:24_1_3"/>
<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</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="ref" href="anproj_1.html#T11" target="_self" title="ANPROJ_1:th.11">Th16</a></span>;<br/><span class="kw">hence </span><a NAME="E4:24_1_3"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp;  not <font color="Olive" title="b1">y</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E2:24_1_3"><span class="lab"><font color="Green" title="E31">A12</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><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E4:24_1"/>
(  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font> or <font color="Maroon" title="c5">p</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  or <font color="Maroon" title="c6">q</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp;  not <font color="Olive" title="b1">y</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  )</span><br/><span class="kw">hence </span><a NAME="E5:24_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp;  not <font color="Olive" title="b1">y</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_1"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="txt" href="#E2:24_1"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="txt" href="#E1:24_1"><span class="lab"><font color="Green" title="E28">A7</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><span class="kw">end;</span></div>
<a NAME="E7:24"/><span class="lab"><font color="Green" title="E28">A13</font></span>: 
 not <font color="Maroon" title="c2">u</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="anproj_1.html#T12" target="_self" title="ANPROJ_1:th.12">Th17</a></span>;<br/>
<a NAME="E8:24"/>
 not  <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="anproj_1.html#T12" target="_self" title="ANPROJ_1:th.12">Th17</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c8">b1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E10:24"/><span class="lab"><font color="Green" title="E29">A14</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E7:24"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="anproj_1.html#T6" target="_self" title="ANPROJ_1:th.6">Th11</a></span>;<br/>
<a NAME="E11:24"/><span class="lab"><font color="Green" title="E30">A15</font></span>: 
 not  <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="anproj_1.html#T12" target="_self" title="ANPROJ_1:th.12">Th17</a></span>;<br/>
<a NAME="E12:24"/>
 not  <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="anproj_1.html#T12" target="_self" title="ANPROJ_1:th.12">Th17</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">a2</font>, <font color="Maroon" title="c10">b2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E14:24"/><span class="lab"><font color="Green" title="E31">A16</font></span>: 
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E5:24"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="ref" href="anproj_1.html#T6" target="_self" title="ANPROJ_1:th.6">Th11</a></span>;<br/>
<a NAME="E15:24"/><span class="lab"><font color="Green" title="E32">A17</font></span>: 
 for <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">d</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">c</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">d</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Olive" title="b2">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c9">a2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</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"><font color="Olive" title="b2">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">c</font>, <font color="Maroon" title="c12">d</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">c</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">d</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c9">a2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</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"><font color="Maroon" title="c12">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">w</font></span>)</span></span><br/>



<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c11">c</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">d</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">q</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">d</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">a2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">b2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:24"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="txt" href="#E14:24"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="txt" href="anproj_1.html#E23"><span class="lab"><font color="Green" title="E21">Lm7</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</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"><font color="Maroon" title="c12">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c9">a2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="anproj_1.html#E23"><span class="lab"><font color="Green" title="E21">Lm7</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c9">a2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</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"><font color="Maroon" title="c12">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D3" title="RLVECT_1:def.3">RLVECT_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c9">a2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</font></span>)</span></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"><font color="Maroon" title="c12">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D3" title="RLVECT_1:def.3">RLVECT_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">c</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c9">a2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</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"><font color="Maroon" title="c12">d</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</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="E33">A18</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:24_3"/><span class="lab"><font color="Green" title="E34">A19</font></span>: 
 not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>
 <span class="kw">and </span><br/><a NAME="E2:24_3"/><span class="lab"><font color="Green" title="E35">A20</font></span>: 
 not <font color="Maroon" title="c5">p</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 <span class="kw">and </span><br/><a NAME="E3:24_3"/><span class="lab"><font color="Green" title="E36">A21</font></span>: 
 not <font color="Maroon" title="c6">q</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 <span class="kw">and </span><br/><a NAME="E4:24_3"/><span class="lab"><font color="Green" title="E37">A22</font></span>: 
<font color="Maroon" title="c8">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp;  not <font color="Olive" title="b1">y</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  )</span><br/><div><span class="lab"><font color="Green" title="E38">A23</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c11">c</font> = 1;<br/><span class="kw">set </span><font color="Maroon" title="c12">d</font> =  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a2</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c13">y</font> = <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">p</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> <span class="p3">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">a2</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">q</font></span>)</span>;<br/><span class="kw">assume </span><a NAME="E1:24_3_1"/><span class="lab"><font color="Green" title="E39">A24</font></span>: 
<font color="Maroon" title="c9">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp;  not <font color="Olive" title="b1">y</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  )</span><br/><a NAME="E2:24_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">a2</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a>  <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="xcmplx_1.html#T202" title="XCMPLX_1:th.202">XCMPLX_1:202</a></span>;<br/><a NAME="E3:24_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A25</font></span>: 
<font color="Maroon" title="c8">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a2</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <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="txt" href="#E4:24_3"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></span>;<br/><a NAME="E4:24_3_1"/><span class="lab"><font color="Green" title="E41">A26</font></span>: 
 not <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">p</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> <span class="p3">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">a2</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">q</font></span>)</span> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:24_3_1_1"/>
<span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">p</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> <span class="p3">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">a2</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">q</font></span>)</span> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<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> = 
<span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">p</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> <span class="p3">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">a2</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D12" title="STRUCT_0:def.12">STRUCT_0:def 12</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">p</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"><font color="Maroon" title="c8">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">a2</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">q</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T24" title="RLVECT_1:th.24">RLVECT_1:24</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">p</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"><span class="p3">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">a2</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">q</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T25" title="RLVECT_1:th.25">RLVECT_1:25</a></span>
;<br/>
<a NAME="E3:24_3_1_1"/><span class="kw">then </span>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">a2</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D10" title="RLVECT_1:def.10">RLVECT_1:def 10</a></span>;<br/>
<a NAME="E4:24_3_1_1"/><span class="kw">then </span>
1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">a2</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T18" title="RLVECT_1:th.18">RLVECT_1:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:24_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_3"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="txt" href="#E3:24_3_1"><span class="lab"><font color="Green" title="E40">A25</font></span></a>, <a class="ref" href="anproj_1.html#D1" target="_self" title="ANPROJ_1:def.1">Def2</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="p1">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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> <span class="p3">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">a2</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c9">a2</font></span>)</span> = 
<font color="Maroon" title="c8">b1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</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> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">a2</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="c9">a2</font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon" title="c8">b1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</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> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_3_1"><span class="lab"><font color="Green" title="E39">A24</font></span></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/><span class="kw">then </span><span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">p</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> <span class="p3">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">a2</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">q</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <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="c3">v</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> <span class="p4">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">a2</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:24"><span class="lab"><font color="Green" title="E32">A17</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">u</font></span>)</span> <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#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> <span class="p4">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">a2</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<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="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">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> <span class="p4">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">a2</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T4" title="RLVECT_1:th.4">RLVECT_1:4</a></span>
;<br/><a NAME="E7:24_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A27</font></span>: 
<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font>,<span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">p</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> <span class="p3">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">a2</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">q</font></span>)</span> <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="#E11:24"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="txt" href="#E7:24"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E5:24"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="ref" href="anproj_1.html#T6" target="_self" title="ANPROJ_1:th.6">Th11</a></span>;<br/><a NAME="E8:24_3_1"/>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">p</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> <span class="p3">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">a2</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">q</font></span>)</span> <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="#E1:24_3"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="txt" href="#E2:24_3"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="txt" href="#E3:24_3"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="ref" href="anproj_1.html#T6" target="_self" title="ANPROJ_1:th.6">Th11</a></span>;<br/><span class="kw">hence </span><a NAME="E9:24_3_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp;  not <font color="Olive" title="b1">y</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_3_1"><span class="lab"><font color="Green" title="E41">A26</font></span></a>, <a class="txt" href="#E7:24_3_1"><span class="lab"><font color="Green" title="E42">A27</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><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3_2"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c11">c</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/><span class="kw">set </span><font color="Maroon" title="c12">d</font> = 1;<br/><span class="kw">set </span><font color="Maroon" title="c13">y</font> = <span class="p1">(<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="c5">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">q</font></span>)</span>;<br/><span class="lab"><font color="Green" title="E39">A28</font></span>: <span class="p1">(<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="c5">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">q</font></span>)</span> = 
<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> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">q</font></span>)</span>
<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="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="c6">q</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c6">q</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T4" title="RLVECT_1:th.4">RLVECT_1:4</a></span>
;<br/><span class="kw">assume </span><a NAME="E2:24_3_2"/>
<font color="Maroon" title="c9">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp;  not <font color="Olive" title="b1">y</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  )</span><br/><a NAME="E3:24_3_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c9">a2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><span class="kw">then </span><span class="p1">(<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="c5">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">q</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <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="c3">v</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">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:24"><span class="lab"><font color="Green" title="E32">A17</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">u</font></span>)</span> <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#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<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="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">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">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T4" title="RLVECT_1:th.4">RLVECT_1:4</a></span>
;<br/><a NAME="E5:24_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A29</font></span>: 
<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font>,<span class="p1">(<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="c5">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">q</font></span>)</span> <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="#E11:24"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="txt" href="#E7:24"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E5:24"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="ref" href="anproj_1.html#T6" target="_self" title="ANPROJ_1:th.6">Th11</a></span>;<br/><a NAME="E6:24_3_2"/>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<span class="p1">(<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="c5">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">q</font></span>)</span> <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="#E1:24_3"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="txt" href="#E2:24_3"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="txt" href="#E3:24_3"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="ref" href="anproj_1.html#T6" target="_self" title="ANPROJ_1:th.6">Th11</a></span>;<br/><span class="kw">hence </span><a NAME="E7:24_3_2"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp;  not <font color="Olive" title="b1">y</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_3"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="txt" href="#E1:24_3_2"><span class="lab"><font color="Green" title="E39">A28</font></span></a>, <a class="txt" href="#E5:24_3_2"><span class="lab"><font color="Green" title="E40">A29</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><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:24_3"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp;  not <font color="Olive" title="b1">y</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_3"><span class="lab"><font color="Green" title="E38">A23</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><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:24_4"/><span class="lab"><font color="Green" title="E34">A30</font></span>: 
 not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>
 <span class="kw">and </span><br/><a NAME="E2:24_4"/><span class="lab"><font color="Green" title="E35">A31</font></span>: 
 not <font color="Maroon" title="c5">p</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 <span class="kw">and </span><br/><a NAME="E3:24_4"/><span class="lab"><font color="Green" title="E36">A32</font></span>: 
 not <font color="Maroon" title="c6">q</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 <span class="kw">and </span><br/><a NAME="E4:24_4"/><span class="lab"><font color="Green" title="E37">A33</font></span>: 
<font color="Maroon" title="c8">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp;  not <font color="Olive" title="b1">y</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c11">c</font> = 1;<br/><span class="kw">set </span><font color="Maroon" title="c12">d</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/><span class="kw">set </span><font color="Maroon" title="c13">y</font> = <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<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="c6">q</font></span>)</span>;<br/><span class="lab"><font color="Green" title="E38">A34</font></span>: <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<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="c6">q</font></span>)</span> = 
<font color="Maroon" title="c5">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<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="c6">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c5">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <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>
<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/>.= 
<font color="Maroon" title="c5">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T4" title="RLVECT_1:th.4">RLVECT_1:4</a></span>
;<br/><a NAME="E2:24_4_1"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c9">a2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_4"><span class="lab"><font color="Green" title="E37">A33</font></span></a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<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="c6">q</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <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="c3">v</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:24"><span class="lab"><font color="Green" title="E32">A17</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">u</font></span>)</span> <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#K3" title="RLVECT_1:func.3">+</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="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<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="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">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="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T4" title="RLVECT_1:th.4">RLVECT_1:4</a></span>
;<br/><a NAME="E4:24_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A35</font></span>: 
<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font>,<span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<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="c6">q</font></span>)</span> <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="#E11:24"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="txt" href="#E7:24"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E5:24"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="ref" href="anproj_1.html#T6" target="_self" title="ANPROJ_1:th.6">Th11</a></span>;<br/><a NAME="E5:24_4_1"/>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<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="c6">q</font></span>)</span> <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="#E1:24_4"><span class="lab"><font color="Green" title="E34">A30</font></span></a>, <a class="txt" href="#E2:24_4"><span class="lab"><font color="Green" title="E35">A31</font></span></a>, <a class="txt" href="#E3:24_4"><span class="lab"><font color="Green" title="E36">A32</font></span></a>, <a class="ref" href="anproj_1.html#T6" target="_self" title="ANPROJ_1:th.6">Th11</a></span>;<br/><span class="kw">hence </span><a NAME="E6:24_4_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp;  not <font color="Olive" title="b1">y</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_4"><span class="lab"><font color="Green" title="E35">A31</font></span></a>, <a class="txt" href="#E1:24_4_1"><span class="lab"><font color="Green" title="E38">A34</font></span></a>, <a class="txt" href="#E4:24_4_1"><span class="lab"><font color="Green" title="E39">A35</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><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:24_4"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp;  not <font color="Olive" title="b1">y</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  )
 ; <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">hence </span><a NAME="E18:24"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<font color="Olive" title="b1">y</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  &amp;  not <font color="Olive" title="b1">y</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E16:24"><span class="lab"><font color="Green" title="E33">A18</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>
