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

<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/>





<b>assume </b><a NAME="E1:26"/><i><font color="Green">E40</font></i>: 
( not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">p</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">q</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  )
 ;<br/>

<a NAME="E2:26"/><b>then </b><i><font color="Green">E41</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> )
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/>
<b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/>



<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_1"/><i><font color="Green">E43</font></i>: 
( not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">v</font> &amp; <font color="Maroon">u</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">v</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  )
 ;<br/><a NAME="E2:26_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">y</font> = <font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font>;<br/><a NAME="E3:26_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> )
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/><b>thus </b><a NAME="E4:26_1"/>
<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a> 
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default">1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default">1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T29">RLVECT_1:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T45">RLVECT_1:45</a></i>
<br/>.= 
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
;<br/><b>hence </b><a NAME="E6:26_1"/>
<font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a> 
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:26_1_2"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font></span>)</span>
 ;<br/><b>then </b><span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/><b>then </b><i><font color="Green">E53</font></i>: <font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">u</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T39">RLVECT_1:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T38">RLVECT_1:38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_1_2_3"/>
<font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:26_1_2_3"/><b>then </b>
<font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>;<br/><b>hence </b><a NAME="E3:26_1_2_3"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#T24">RLVECT_1:24</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:26_1_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:26_1"/>
not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:26_1_3"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font></span>)</span>
 ;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/><b>then </b><i><font color="Green">E58</font></i>: <font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T39">RLVECT_1:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T38">RLVECT_1:38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_1_3_3"/>
<font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:26_1_3_3"/><b>then </b>
<font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>;<br/><b>hence </b><a NAME="E3:26_1_3_3"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#T24">RLVECT_1:24</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:26_1_3"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:26_1"/>
not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">v</font>,<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_2"/><i><font color="Green">E60</font></i>: 
 <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
 ;<br/><a NAME="E2:26_2"/><b>then </b><i><font color="Green">E61</font></i>: 
( not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">p</font> implies ( <font color="Maroon">p</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">p</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a>  &amp; not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">p</font> &amp; not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">v</font>,<font color="Maroon">p</font> ) )
 <b>by </b><i>, , </i>;<br/><a NAME="E3:26_2"/>
( not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">q</font> implies ( <font color="Maroon">q</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">q</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a>  &amp; not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">q</font> &amp; not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">v</font>,<font color="Maroon">q</font> ) )
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E4:26_2"/>
 ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">y</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Olive">y</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a>  &amp; not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u</font>,<font color="Olive">y</font> &amp; not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">v</font>,<font color="Olive">y</font> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_3"/>
( not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">v</font> &amp; not <font color="Maroon">u</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  )
 ;<br/><a NAME="E2:26_3"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/><a NAME="E3:26_3"/><b>then </b><i><font color="Green">E69</font></i>: 
( not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">v</font>,<font color="Maroon">p</font> implies ( not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">v</font>,<font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">p</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a>  &amp; not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">p</font> ) )
 <b>by </b><i>, , , </i>;<br/><a NAME="E4:26_3"/>
( not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">v</font>,<font color="Maroon">q</font> implies ( not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">v</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">q</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a>  &amp; not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">q</font> ) )
 <b>by </b><i>, , , , </i>;<br/><b>hence </b><a NAME="E5:26_3"/>
 ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">y</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Olive">y</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a>  &amp; not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u</font>,<font color="Olive">y</font> &amp; not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">v</font>,<font color="Olive">y</font> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_4"/>
( not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">v</font> &amp; not <font color="Maroon">v</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  )
 ;<br/><a NAME="E2:26_4"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/><a NAME="E3:26_4"/><b>then </b><i><font color="Green">E71</font></i>: 
( not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">p</font> implies ( <font color="Maroon">p</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">p</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a>  &amp; not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">p</font> &amp; not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">v</font>,<font color="Maroon">p</font> ) )
 <b>by </b><i>, , , </i>;<br/><a NAME="E4:26_4"/>
( not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">q</font> implies ( <font color="Maroon">q</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">q</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a>  &amp; not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">q</font> &amp; not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">v</font>,<font color="Maroon">q</font> ) )
 <b>by </b><i>, , , , </i>;<br/><b>hence </b><a NAME="E5:26_4"/>
 ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">y</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Olive">y</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a>  &amp; not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u</font>,<font color="Olive">y</font> &amp; not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">v</font>,<font color="Olive">y</font> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:26"/>
 ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">y</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Olive">y</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a>  &amp; not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u</font>,<font color="Olive">y</font> &amp; not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">v</font>,<font color="Olive">y</font> )
 <b>by </b><i>, , </i>;<br/>


</div>
