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<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">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/><b>let </b><font color="Maroon">w</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:23"/><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>
 <b>and </b><br/><a NAME="E2:23"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">u</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a> 
 <b>and </b><br/><a NAME="E3:23"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">v</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a> 
 <b>and </b><br/><a NAME="E4:23"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">w</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a> 
 <b>and </b><br/><a NAME="E5:23"/><i><font color="Green">E45</font></i>: 
( <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/>

<b>consider </b><font color="Maroon">a1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:23"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <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>
 <b>by </b><i>, , , </i>;<br/>
<b>consider </b><font color="Maroon">a2</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E9:23"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">v</font> <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>
 <b>by </b><i>, , , </i>;<br/>
<b>consider </b><font color="Maroon">a3</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b3</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E11:23"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a3</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">b3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, , , </i>;<br/>
<b>set </b><font color="Maroon">a</font> = <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">b</font> = <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c</font> = <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span>;<br/>
<a NAME="E12:23"/><i><font color="Green">E59</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a3</font></span>)</span></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#T23">RLVECT_1:23</a></i>;<br/>
<a NAME="E13:23"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span></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#T23">RLVECT_1:23</a></i>;<br/>
<a NAME="E14:23"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a3</font></span>)</span></span>)</span> <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/>
<a NAME="E15:23"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a3</font></span>)</span></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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font></span>)</span> <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <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">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E16:23"/><b>then </b>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</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="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> <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="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E17:23"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span> <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="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a2</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> <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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a3</font></span>)</span> <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E18:23"/><i><font color="Green">E62</font></i>: 
( <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">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font></span>)</span> <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<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="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> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a2</font></span>)</span> <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> <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"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</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"><font color="Maroon">b2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span> &amp; <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">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a3</font></span>)</span> <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a3</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">b3</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span> )
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1"/><i><font color="Green">E63</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><a NAME="E2:23_1"/><b>then </b><i><font color="Green">E69</font></i>: 
0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span>
 ;<br/><a NAME="E3:23_1"/><i><font color="Green">E70</font></i>: 
(  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">w</font>,<font color="Maroon">u</font> &amp;  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">v</font> &amp;  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">v</font>,<font color="Maroon">w</font> implies <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">w</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a>  )
 <b>by </b><i/>;<br/><a NAME="E4:23_1"/><i><font color="Green">E71</font></i>: 
( ( ( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> &amp;  <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">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> ) or ( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> &amp;  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">v</font>,<font color="Maroon">w</font> ) or (  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">w</font>,<font color="Maroon">u</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> ) ) implies <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">w</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a>  )
 <b>by </b><i/>;<br/><a NAME="E5:23_1"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> &amp; (  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">v</font>,<font color="Maroon">w</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> or <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> ) implies <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">w</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a>  )
 <b>by </b><i/>;<br/><a NAME="E6:23_1"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> &amp; (  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">v</font>,<font color="Maroon">w</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> or <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> ) implies <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">w</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a>  )
 <b>by </b><i/>;<br/><a NAME="E7:23_1"/>
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> &amp; (  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">v</font>,<font color="Maroon">w</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> or <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> ) implies <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">w</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a>  )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:23_1"/>
<font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">w</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a> 
 <b>by </b><i>, , , , , , , , , , <a class="ref" href="anproj_1.html#D2" target="_self">Def2</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E20:23"/>
<font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">w</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a> 
 <b>by </b><i>, , , , , </i>;<br/>


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