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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">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">q</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>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<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">r</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="E1:30"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">q</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a>  &amp; <font color="Maroon">w</font>,<font color="Maroon">y</font>,<font color="Maroon">q</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a>  &amp; <font color="Maroon">u</font>,<font color="Maroon">w</font>,<font color="Maroon">p</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a>  &amp; <font color="Maroon">v</font>,<font color="Maroon">y</font>,<font color="Maroon">p</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a>  &amp; <font color="Maroon">u</font>,<font color="Maroon">y</font>,<font color="Maroon">r</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a>  &amp; <font color="Maroon">v</font>,<font color="Maroon">w</font>,<font color="Maroon">r</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a>  )
 <b>and </b><br/><a NAME="E2:30"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a> 
 <b>and </b><br/><a NAME="E3:30"/><i><font color="Green">E42</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>  &amp; <font color="Maroon">r</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  )
 ;<br/>

<b>assume </b><a NAME="E4:30"/><i><font color="Green">E43</font></i>: 
( not <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">y</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a>  &amp; not <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>  &amp; not <font color="Maroon">u</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a>  &amp; not <font color="Maroon">v</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font> <a href="anproj_1.html#R2" target="_self">are_LinDep</a>  )
 ;<br/>

<a NAME="E5:30"/><b>then </b><i><font color="Green">E45</font></i>: 
( <font color="Maroon">u</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">v</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">w</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  &amp; <font color="Maroon">y</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E6:30"/><i><font color="Green">E52</font></i>: 
( not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">v</font> &amp; not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">v</font>,<font color="Maroon">y</font> &amp; not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">y</font>,<font color="Maroon">u</font> &amp; not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">v</font>,<font color="Maroon">w</font> &amp; not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">w</font>,<font color="Maroon">y</font> &amp; not  <a href="anproj_1.html#R1" target="_self">are_Prop</a> <font color="Maroon">u</font>,<font color="Maroon">w</font> )
 
<b>by </b><i>, </i>;<br/>
<b>then 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="E8:30"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">p</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">u</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">w</font></span>)</span>
 <b>by </b><i>, , </i>;<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="E10:30"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">p</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">v</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">y</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="E12:30"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">q</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">u</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">v</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="E14:30"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">q</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">w</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">y</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="E16:30"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">r</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">u</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">y</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="E18:30"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">r</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">v</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">w</font></span>)</span>
 <b>by </b><i>, , , </i>;<br/>
<b>consider </b><font color="Maroon">A</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">B</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">C</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E20:30"/><i><font color="Green">E63</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</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">B</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> &amp; ( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) )
 <b>by </b><i>, </i>;<br/>
<a NAME="E21:30"/>
 <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"><font color="Maroon">A</font> <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"><font color="Maroon">C</font> <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">u</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"><font color="Maroon">A</font> <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"><font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a2'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</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"><font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2'</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <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">y</font></span>)</span>
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E22:30"/><b>then </b><i><font color="Green">E69</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a3</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a2'</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2'</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E23:30"/>
 <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"><font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">C</font> <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">u</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"><font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a1'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <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"><font color="Maroon">C</font> <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">y</font></span>)</span>
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E24:30"/><b>then </b><i><font color="Green">E70</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a3</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a1'</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1'</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E25:30"/>
 <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"><font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">C</font> <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">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3'</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span>
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E26:30"/><b>then </b><i><font color="Green">E71</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a3'</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3'</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, </i>;<br/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a3'</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><font color="Maroon">b3'</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="rlvect_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a2'</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><font color="Maroon">b2'</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a1'</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">b1'</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , , , <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a3'</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a1'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3'</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a2'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</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"><font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2'</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E28:30"/><b>then </b><i><font color="Green">E72</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a3'</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a1'</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3'</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a2'</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2'</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1'</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, </i>;<br/>
<div><i><font color="Green">E78</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:30_2"/>
( <font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a3</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font></span>)</span> &amp; <font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a3</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a2</font></span>)</span> &amp; <font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a2</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="anproj_1.html#D2" target="_self">Def2</a>, <a class="ref" href="anproj_1.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E3:30_2"/><b>then </b>
<font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E4:30_2"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="anproj_1.html#E3:30"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/><a NAME="E5:30_2"/>
( <font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a2'</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font></span>)</span> &amp; <font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a2'</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3'</font></span>)</span> &amp; <font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3'</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="anproj_1.html#T5" target="_self">Th5</a>, <a class="txt" href="anproj_1.html#E1:30"><i><font color="Green">E40</font></i></a></i>;<br/><a NAME="E6:30_2"/><b>then </b>
<font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>then </b><font color="Maroon">p</font> = 
<span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i>, <a class="txt" href="anproj_1.html#E3:30"><i><font color="Green">E42</font></i></a>, <a class="txt" href="anproj_1.html#E4:30"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/><b>hence </b><a NAME="E8:30_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E82</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_3"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:30_3"/>
( <font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a3</font></span>)</span> &amp; <font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a2</font></span>)</span> &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a3</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a2</font> )
 <b>by </b><i>, <a class="ref" href="anproj_1.html#D2" target="_self">Def2</a>, <a class="ref" href="anproj_1.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E3:30_3"/><b>then </b>
<font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E4:30_3"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="anproj_1.html#E4:30"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/><a NAME="E5:30_3"/>
( <font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a1'</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font></span>)</span> &amp; <font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a1'</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a3'</font></span>)</span> &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a3'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="anproj_1.html#D2" target="_self">Def2</a>, <a class="ref" href="anproj_1.html#T5" target="_self">Th5</a>, <a class="txt" href="anproj_1.html#E1:30"><i><font color="Green">E40</font></i></a></i>;<br/><a NAME="E6:30_3"/><b>then </b>
<font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>then </b><font color="Maroon">q</font> = 
<span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i>, <a class="txt" href="anproj_1.html#E4:30"><i><font color="Green">E43</font></i></a>, <a class="ref" href="anproj_1.html#T6" target="_self">Th6</a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/><b>hence </b><a NAME="E8:30_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_4"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:30_4"/>
( <font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a3</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a2</font></span>)</span> &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a3</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="anproj_1.html#D2" target="_self">Def2</a>, <a class="ref" href="anproj_1.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E3:30_4"/><b>then </b>
<font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E4:30_4"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="anproj_1.html#E4:30"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/><a NAME="E5:30_4"/>
( <font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2'</font></span>)</span> &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b2'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1'</font> &amp; <font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1'</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="anproj_1.html#D2" target="_self">Def2</a>, <a class="txt" href="anproj_1.html#E1:30"><i><font color="Green">E40</font></i></a></i>;<br/><a NAME="E6:30_4"/><b>then </b>
<font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b3</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>then </b><font color="Maroon">r</font> = 
<span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="txt" href="anproj_1.html#E4:30"><i><font color="Green">E43</font></i></a>, <a class="ref" href="anproj_1.html#T6" target="_self">Th6</a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default">0 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/><b>hence </b><a NAME="E8:30_4"/>
contradiction
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E32:30"/>
contradiction
 <b>by </b><i>, <a class="txt" href="anproj_1.html#E2:30"><i><font color="Green">E41</font></i></a>, <a class="txt" href="anproj_1.html#E3:30"><i><font color="Green">E42</font></i></a></i>;<br/>


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