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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>;<br/><b>let </b><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>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:25"/><i><font color="Green" title="E22">A1</font></i>: 
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> 
 <b>and </b><br/><a NAME="E2:25"/><i><font color="Green" title="E23">A2</font></i>: 
<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> 
 <b>and </b><br/><a NAME="E3:25"/><i><font color="Green" title="E24">A3</font></i>: 
<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> 
 ;<br/>

<a NAME="E4:25"/><i><font color="Green" title="E25">A4</font></i>: 
( 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> &amp; 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> &amp; 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> &amp; <font color="Maroon" title="c2">u</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c3">v</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c4">w</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  )
 
<b>by </b><i><a class="txt" href="anproj_1.html#E1:25"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="anproj_1.html#T17" target="_self" title="ANPROJ_1:th.17">Th17</a></i>;<br/>
<b>then consider </b><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><b> such that </b><br/><a NAME="E6:25"/><i><font color="Green" title="E26">A5</font></i>: 
<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#K2" title="RLVECT_1:func.2">*</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#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c3">v</font></span>)</span>
 <b>by </b><i><a class="txt" href="anproj_1.html#E2:25"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="anproj_1.html#T11" target="_self" title="ANPROJ_1:th.11">Th11</a></i>;<br/>
<b>consider </b><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><b> such that </b><br/><a NAME="E8:25"/><i><font color="Green" title="E27">A6</font></i>: 
<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#K2" title="RLVECT_1:func.2">*</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#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
 <b>by </b><i><a class="txt" href="anproj_1.html#E3:25"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="anproj_1.html#E4:25"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="anproj_1.html#T11" target="_self" title="ANPROJ_1:th.11">Th11</a></i>;<br/>
<a NAME="E9:25"/><i><font color="Green" title="E28">A7</font></i>: 
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#K2" title="RLVECT_1:func.2">*</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#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p4">(<span class="default"><font color="Olive" title="b2">d</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c9">a2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>let </b><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>;<br/>



<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c11">c</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K2" title="RLVECT_1:func.2">*</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#K2" title="RLVECT_1:func.2">*</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#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="anproj_1.html#E6:25"><i><font color="Green" title="E26">A5</font></i></a>, <a class="txt" href="anproj_1.html#E8:25"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="anproj_1.html#E29"><i><font color="Green" title="E21">Lm7</font></i></a></i>
<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#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c9">a2</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="anproj_1.html#E29"><i><font color="Green" title="E21">Lm7</font></i></a></i>
<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#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c9">a2</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c9">a2</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">c</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">d</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c9">a2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E29">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_2"/><i><font color="Green" title="E30">A9</font></i>: 
(  <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  not <font color="Maroon" title="c5">p</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  or  not <font color="Maroon" title="c6">q</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  )
 ;<br/><div><i><font color="Green" title="E31">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_2_1"/><i><font color="Green" title="E32">A11</font></i>: 
 <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>
 ;<br/><a NAME="E2:25_2_1"/><i><font color="Green" title="E33">A12</font></i>: 
<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> 
 <b>by </b><i><a class="ref" href="anproj_1.html#T16" target="_self" title="ANPROJ_1:th.16">Th16</a></i>;<br/><a NAME="E3:25_2_1"/>
<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> 
 <b>by </b><i><a class="txt" href="anproj_1.html#E1:25_2_1"><i><font color="Green" title="E32">A11</font></i></a>, <a class="ref" href="anproj_1.html#T16" target="_self" title="ANPROJ_1:th.16">Th16</a></i>;<br/><b>hence </b><a NAME="E4:25_2_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; <font color="Olive" title="b1">y</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  )
 <b>by </b><i><a class="txt" href="anproj_1.html#E4:25"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="anproj_1.html#E2:25_2_1"><i><font color="Green" title="E33">A12</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E32">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_2_2"/>
not <font color="Maroon" title="c5">p</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a> 
 ;<br/><a NAME="E2:25_2_2"/><b>then </b><i><font color="Green" title="E33">A14</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D13" title="RLVECT_1:def.13">RLVECT_1:def 13</a></i>;<br/><a NAME="E3:25_2_2"/><i><font color="Green" title="E34">A15</font></i>: 
<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> 
 <b>by </b><i><a class="ref" href="anproj_1.html#T16" target="_self" title="ANPROJ_1:th.16">Th16</a></i>;<br/><a NAME="E4:25_2_2"/>
<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> 
 <b>by </b><i><a class="txt" href="anproj_1.html#E2:25_2_2"><i><font color="Green" title="E33">A14</font></i></a>, <a class="ref" href="anproj_1.html#T15" target="_self" title="ANPROJ_1:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E5:25_2_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; <font color="Olive" title="b1">y</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  )
 <b>by </b><i><a class="txt" href="anproj_1.html#E4:25"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="anproj_1.html#E3:25_2_2"><i><font color="Green" title="E34">A15</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_2_3"/>
not <font color="Maroon" title="c6">q</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a> 
 ;<br/><a NAME="E2:25_2_3"/><b>then </b><i><font color="Green" title="E33">A16</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D13" title="RLVECT_1:def.13">RLVECT_1:def 13</a></i>;<br/><a NAME="E3:25_2_3"/><i><font color="Green" title="E34">A17</font></i>: 
<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> 
 <b>by </b><i><a class="ref" href="anproj_1.html#T16" target="_self" title="ANPROJ_1:th.16">Th16</a></i>;<br/><a NAME="E4:25_2_3"/>
<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> 
 <b>by </b><i><a class="txt" href="anproj_1.html#E2:25_2_3"><i><font color="Green" title="E33">A16</font></i></a>, <a class="ref" href="anproj_1.html#T15" target="_self" title="ANPROJ_1:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E5:25_2_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; <font color="Olive" title="b1">y</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  )
 <b>by </b><i><a class="txt" href="anproj_1.html#E4:25"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="anproj_1.html#E3:25_2_3"><i><font color="Green" title="E34">A17</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:25_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; <font color="Olive" title="b1">y</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  )
 <b>by </b><i><a class="txt" href="anproj_1.html#E1:25_2"><i><font color="Green" title="E30">A9</font></i></a>, <a class="txt" href="anproj_1.html#E2:25_2"><i><font color="Green" title="E31">A10</font></i></a>, <a class="txt" href="anproj_1.html#E3:25_2"><i><font color="Green" title="E32">A13</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E30">A18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_3"/><i><font color="Green" title="E31">A19</font></i>: 
( 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> &amp; <font color="Maroon" title="c5">p</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c6">q</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c8">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3_1"><b>now </b></a><div class="add"><b>set </b><font color="Maroon" title="c11">c</font> = 1;<br/><b>set </b><font color="Maroon" title="c12">d</font> = 0;<br/><b>set </b><font color="Maroon" title="c13">y</font> = <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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">0 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">q</font></span>)</span>;<br/><a NAME="E1:25_3_1"/><i><font color="Green" title="E32">A20</font></i>: 
<span class="p1">(<span class="default">1 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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">0 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">q</font></span>)</span> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3_1_1"><b>proof </b></a><div class="add">

<span class="p1">(<span class="default">1 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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">0 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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">0 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">q</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
<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>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23" title="RLVECT_1:th.23">RLVECT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c5">p</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
;<br/>
<b>hence </b><a NAME="E2:25_3_1_1"/>
<span class="p1">(<span class="default">1 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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">0 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">q</font></span>)</span> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a> 
 <b>by </b><i><a class="txt" href="anproj_1.html#E1:25_3"><i><font color="Green" title="E31">A19</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E2:25_3_1"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default">0 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c9">a2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="anproj_1.html#E1:25_3"><i><font color="Green" title="E31">A19</font></i></a></i>;<br/><b>then </b><span class="p1">(<span class="default">1 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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">0 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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">0 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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">0 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<b>by </b><i><a class="txt" href="anproj_1.html#E9:25"><i><font color="Green" title="E28">A7</font></i></a></i>
<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#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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">0 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23" title="RLVECT_1:th.23">RLVECT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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">0 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
;<br/><a NAME="E4:25_3_1"/><b>then </b><i><font color="Green" title="E33">A21</font></i>: 
<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#K2" title="RLVECT_1:func.2">*</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">0 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">q</font></span>)</span> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_1.html#E4:25"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="anproj_1.html#T11" target="_self" title="ANPROJ_1:th.11">Th11</a></i>;<br/><a NAME="E5:25_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#K2" title="RLVECT_1:func.2">*</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">0 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">q</font></span>)</span> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_1.html#E1:25_3"><i><font color="Green" title="E31">A19</font></i></a>, <a class="ref" href="anproj_1.html#T11" target="_self" title="ANPROJ_1:th.11">Th11</a></i>;<br/><b>hence </b><a NAME="E6:25_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; <font color="Olive" title="b1">y</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  )
 <b>by </b><i><a class="txt" href="anproj_1.html#E1:25_3_1"><i><font color="Green" title="E32">A20</font></i></a>, <a class="txt" href="anproj_1.html#E4:25_3_1"><i><font color="Green" title="E33">A21</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:25_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; <font color="Olive" title="b1">y</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  )
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_4"/><i><font color="Green" title="E31">A22</font></i>: 
( 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> &amp; <font color="Maroon" title="c5">p</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c6">q</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c8">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 ;<br/><div><i><font color="Green" title="E32">A23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_4_1"/><i><font color="Green" title="E33">A24</font></i>: 
<font color="Maroon" title="c9">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><b>set </b><font color="Maroon" title="c11">c</font> = 0;<br/><b>set </b><font color="Maroon" title="c12">d</font> = 1;<br/><b>set </b><font color="Maroon" title="c13">y</font> = <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">q</font></span>)</span>;<br/><a NAME="E2:25_4_1"/><i><font color="Green" title="E34">A25</font></i>: 
<span class="p1">(<span class="default">0 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">q</font></span>)</span> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_4_1_1"><b>proof </b></a><div class="add">

<span class="p1">(<span class="default">0 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K2" title="RLVECT_1:func.2">*</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#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">q</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23" title="RLVECT_1:th.23">RLVECT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">q</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c6">q</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
;<br/>
<b>hence </b><a NAME="E2:25_4_1_1"/>
<span class="p1">(<span class="default">0 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">q</font></span>)</span> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a> 
 <b>by </b><i><a class="txt" href="anproj_1.html#E1:25_4"><i><font color="Green" title="E31">A22</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:25_4_1"/>
<span class="p1">(<span class="default">0 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c9">a2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="anproj_1.html#E1:25_4_1"><i><font color="Green" title="E33">A24</font></i></a></i>;<br/><b>then </b><span class="p1">(<span class="default">0 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K2" title="RLVECT_1:func.2">*</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">0 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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">0 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<b>by </b><i><a class="txt" href="anproj_1.html#E9:25"><i><font color="Green" title="E28">A7</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">0 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23" title="RLVECT_1:th.23">RLVECT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
;<br/><a NAME="E5:25_4_1"/><b>then </b><i><font color="Green" title="E35">A26</font></i>: 
<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">w</font>,<span class="p1">(<span class="default">0 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">q</font></span>)</span> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_1.html#E4:25"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="anproj_1.html#T11" target="_self" title="ANPROJ_1:th.11">Th11</a></i>;<br/><a NAME="E6:25_4_1"/>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>,<span class="p1">(<span class="default">0 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">q</font></span>)</span> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_1.html#E1:25_4"><i><font color="Green" title="E31">A22</font></i></a>, <a class="ref" href="anproj_1.html#T11" target="_self" title="ANPROJ_1:th.11">Th11</a></i>;<br/><b>hence </b><a NAME="E7:25_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; <font color="Olive" title="b1">y</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  )
 <b>by </b><i><a class="txt" href="anproj_1.html#E2:25_4_1"><i><font color="Green" title="E34">A25</font></i></a>, <a class="txt" href="anproj_1.html#E5:25_4_1"><i><font color="Green" title="E35">A26</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_4_2"/><i><font color="Green" title="E33">A27</font></i>: 
<font color="Maroon" title="c9">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><b>set </b><font color="Maroon" title="c11">c</font> = 1;<br/><b>set </b><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#K4" title="REAL_1:func.4">*</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/><b>set </b><font color="Maroon" title="c13">y</font> = <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</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#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">q</font></span>)</span>;<br/><a NAME="E2:25_4_2"/>
<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> 0
 <b>by </b><i><a class="txt" href="anproj_1.html#E1:25_4_2"><i><font color="Green" title="E33">A27</font></i></a>, <a class="ref" href="xcmplx_1.html#T203" title="XCMPLX_1:th.203">XCMPLX_1:203</a></i>;<br/><a NAME="E3:25_4_2"/><b>then </b><i><font color="Green" title="E34">A28</font></i>: 
<font color="Maroon" title="c8">b1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</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> 0
 <b>by </b><i><a class="txt" href="anproj_1.html#E1:25_4"><i><font color="Green" title="E31">A22</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>;<br/><a NAME="E4:25_4_2"/><i><font color="Green" title="E35">A29</font></i>: 
<span class="p1">(<span class="default">1 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</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#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">q</font></span>)</span> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_4_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_4_2_1"/>
not <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</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#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">q</font></span>)</span> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a> 
 ;<br/>

<b>then </b> <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#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</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#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">q</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D13" title="RLVECT_1:def.13">RLVECT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</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#K2" title="RLVECT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c6">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T38" title="RLVECT_1:th.38">RLVECT_1:38</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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="rlvect_1.html#K4" title="RLVECT_1: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#K4" title="REAL_1:func.4">*</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#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T39" title="RLVECT_1:th.39">RLVECT_1:39</a></i>
;<br/>
<a NAME="E3:25_4_2_1"/><b>then </b>
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1: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#K4" title="REAL_1:func.4">*</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#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D10" title="RLVECT_1:def.10">RLVECT_1:def 10</a></i>;<br/>
<a NAME="E4:25_4_2_1"/><b>then </b>
1 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</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#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">q</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T31" title="RLVECT_1:th.31">RLVECT_1:31</a></i>;<br/>
<b>hence </b><a NAME="E5:25_4_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="anproj_1.html#E1:25_4"><i><font color="Green" title="E31">A22</font></i></a>, <a class="txt" href="anproj_1.html#E3:25_4_2"><i><font color="Green" title="E34">A28</font></i></a>, <a class="ref" href="anproj_1.html#D2" target="_self" title="ANPROJ_1:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div><span class="p1">(<span class="default">1 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="real_1.html#K3" title="REAL_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#K4" title="REAL_1:func.4">*</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#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c9">a2</font></span>)</span> = 
<font color="Maroon" title="c8">b1</font> <a href="real_1.html#K3" title="REAL_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> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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#K4" title="REAL_1:func.4">*</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#K3" title="REAL_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> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="anproj_1.html#E1:25_4_2"><i><font color="Green" title="E33">A27</font></i></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>
<br/>.= 
0

;<br/><b>then </b><span class="p1">(<span class="default">1 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</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#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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">0 <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</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#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<b>by </b><i><a class="txt" href="anproj_1.html#E9:25"><i><font color="Green" title="E28">A7</font></i></a></i>
<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#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</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#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23" title="RLVECT_1:th.23">RLVECT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">a1</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</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#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
;<br/><a NAME="E7:25_4_2"/><b>then </b><i><font color="Green" title="E36">A30</font></i>: 
<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#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</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#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">q</font></span>)</span> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_1.html#E4:25"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="anproj_1.html#T11" target="_self" title="ANPROJ_1:th.11">Th11</a></i>;<br/><a NAME="E8:25_4_2"/>
<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#K2" title="RLVECT_1:func.2">*</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#K4" title="REAL_1:func.4">*</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#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c6">q</font></span>)</span> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_1.html#E1:25_4"><i><font color="Green" title="E31">A22</font></i></a>, <a class="ref" href="anproj_1.html#T11" target="_self" title="ANPROJ_1:th.11">Th11</a></i>;<br/><b>hence </b><a NAME="E9:25_4_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; <font color="Olive" title="b1">y</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  )
 <b>by </b><i><a class="txt" href="anproj_1.html#E4:25_4_2"><i><font color="Green" title="E35">A29</font></i></a>, <a class="txt" href="anproj_1.html#E7:25_4_2"><i><font color="Green" title="E36">A30</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:25_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; <font color="Olive" title="b1">y</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  )
 <b>by </b><i><a class="txt" href="anproj_1.html#E2:25_4"><i><font color="Green" title="E32">A23</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:25"/>
 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; <font color="Olive" title="b1">y</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  )
 <b>by </b><i><a class="txt" href="anproj_1.html#E10:25"><i><font color="Green" title="E29">A8</font></i></a>, <a class="txt" href="anproj_1.html#E11:25"><i><font color="Green" title="E30">A18</font></i></a></i>;<br/>


</div>
