<?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>let </b><font color="Maroon">a1</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">b1</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">a2</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">b2</font> be   <a href="real_1.html#NM1">Real</a>;<br/>













<b>assume </b><a NAME="E1:15"/><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; <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font></span>)</span> &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="E2:15"/><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/>
<a NAME="E3:15"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">b2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</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">b1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</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">b1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</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">b1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</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">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</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">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<b>then </b><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</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">b1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</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"><span class="p2">(<span class="default"><font color="Maroon">b2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T49">RLVECT_1:49</a></i>
;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</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">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</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">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</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">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</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"><span class="p2">(<span class="default"><font color="Maroon">b2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</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/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T49">RLVECT_1:49</a></i>
;<br/><br/>


</div><b>end;</b></div>
<div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">a1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:15_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</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/><a NAME="E3:15_2"/><b>then </b>
<font color="Maroon">b2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b1</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/><b>hence </b><a NAME="E4:15_2"/>
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_3"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">b2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:15_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</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/><a NAME="E3:15_3"/><b>then </b>
<font color="Maroon">a1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a2</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/><b>hence </b><a NAME="E4:15_3"/>
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:15"/>
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> )
 <b>by </b><i>, , , </i>;<br/>


</div>
