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<div class="add">

<b>let </b><font color="Maroon" title="c1">V</font> be  <a href="anproj_2.html#V1" title="ANPROJ_2:attr.1">up-3-dimensional</a> <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="anproj_1.html#K6" title="ANPROJ_1:func.6">ProjectiveSpace</a> <font color="Maroon" title="c1">V</font> is <a href="anproj_2.html#V6" title="ANPROJ_2:attr.6">Pappian</a></span><br/><b>let </b><font color="Maroon" title="c2">o</font>, <font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">p2</font>, <font color="Maroon" title="c5">p3</font>, <font color="Maroon" title="c6">q1</font>, <font color="Maroon" title="c7">q2</font>, <font color="Maroon" title="c8">q3</font>, <font color="Maroon" title="c9">r1</font>, <font color="Maroon" title="c10">r2</font>, <font color="Maroon" title="c11">r3</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K6" title="ANPROJ_1:func.6">ProjectiveSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="anproj_2.html#D13" target="_self" title="ANPROJ_2:def.13">ANPROJ_2:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p2</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">p3</font> &amp; <font color="Maroon" title="c4">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">p3</font> &amp; <font color="Maroon" title="c3">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p2</font> &amp; <font color="Maroon" title="c3">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">p3</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">q2</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">q3</font> &amp; <font color="Maroon" title="c7">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">q3</font> &amp; <font color="Maroon" title="c6">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">q2</font> &amp; <font color="Maroon" title="c6">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">q3</font> &amp; not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c6">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c8">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c7">q2</font>,<font color="Maroon" title="c11">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c11">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c8">q3</font>,<font color="Maroon" title="c10">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">p3</font>,<font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c10">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c8">q3</font>,<font color="Maroon" title="c9">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">p3</font>,<font color="Maroon" title="c7">q2</font>,<font color="Maroon" title="c9">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  implies <font color="Maroon" title="c9">r1</font>,<font color="Maroon" title="c10">r2</font>,<font color="Maroon" title="c11">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:88"/><i><font color="Green" title="E79">A1</font></i>: 
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p2</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">p3</font> &amp; <font color="Maroon" title="c4">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">p3</font> &amp; <font color="Maroon" title="c3">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p2</font> &amp; <font color="Maroon" title="c3">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">p3</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">q2</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">q3</font> &amp; <font color="Maroon" title="c7">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">q3</font> &amp; <font color="Maroon" title="c6">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">q2</font> &amp; <font color="Maroon" title="c6">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">q3</font> )
 <b>and </b><br/><a NAME="E2:88"/><i><font color="Green" title="E80">A2</font></i>: 
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c6">q1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>and </b><br/><a NAME="E3:88"/><i><font color="Green" title="E81">A3</font></i>: 
( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c8">q3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>and </b><br/><a NAME="E4:88"/><i><font color="Green" title="E82">A4</font></i>: 
( <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c7">q2</font>,<font color="Maroon" title="c11">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c11">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c8">q3</font>,<font color="Maroon" title="c10">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">p3</font>,<font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c10">r2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c8">q3</font>,<font color="Maroon" title="c9">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c5">p3</font>,<font color="Maroon" title="c7">q2</font>,<font color="Maroon" title="c9">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">r1</font>,<font color="Maroon" title="c10">r2</font>,<font color="Maroon" title="c11">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/>

<b>consider </b><font color="Maroon" title="c12">o'</font>, <font color="Maroon" title="c13">p1'</font>, <font color="Maroon" title="c14">p2'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E6:88"/><i><font color="Green" title="E83">A5</font></i>: 
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c12">o'</font> &amp; <font color="Maroon" title="c3">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c13">p1'</font> &amp; <font color="Maroon" title="c4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c14">p2'</font> &amp; <font color="Maroon" title="c12">o'</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c13">p1'</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c14">p2'</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c12">o'</font>,<font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c14">p2'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:88"><i><font color="Green" title="E81">A3</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">o''</font>, <font color="Maroon" title="c16">p1''</font>, <font color="Maroon" title="c17">p3'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E8:88"/><i><font color="Green" title="E84">A6</font></i>: 
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c15">o''</font> &amp; <font color="Maroon" title="c3">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c16">p1''</font> &amp; <font color="Maroon" title="c5">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c17">p3'</font> &amp; <font color="Maroon" title="c15">o''</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c16">p1''</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c17">p3'</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c15">o''</font>,<font color="Maroon" title="c16">p1''</font>,<font color="Maroon" title="c17">p3'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:88"><i><font color="Green" title="E81">A3</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></i>;<br/>
<a NAME="E9:88"/><i><font color="Green" title="E85">A7</font></i>: 
<font color="Maroon" title="c12">o'</font>,<font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c17">p3'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><b>proof </b></a><div class="add">

<a NAME="E1:88_1"/>
(  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c15">o''</font>,<font color="Maroon" title="c12">o'</font> &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c16">p1''</font>,<font color="Maroon" title="c13">p1'</font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E6:88"><i><font color="Green" title="E83">A5</font></i></a>, <a class="txt" href="anproj_2.html#E8:88"><i><font color="Green" title="E84">A6</font></i></a>, <a class="ref" href="anproj_1.html#T35" title="ANPROJ_1:th.35">ANPROJ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:88_1"/>
<font color="Maroon" title="c12">o'</font>,<font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c17">p3'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E8:88"><i><font color="Green" title="E84">A6</font></i></a>, <a class="ref" href="anproj_1.html#T9" title="ANPROJ_1:th.9">ANPROJ_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c18">o'''</font>, <font color="Maroon" title="c19">q1'</font>, <font color="Maroon" title="c20">q2'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E11:88"/><i><font color="Green" title="E86">A8</font></i>: 
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c18">o'''</font> &amp; <font color="Maroon" title="c6">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c19">q1'</font> &amp; <font color="Maroon" title="c7">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c20">q2'</font> &amp; <font color="Maroon" title="c18">o'''</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c19">q1'</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c20">q2'</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c18">o'''</font>,<font color="Maroon" title="c19">q1'</font>,<font color="Maroon" title="c20">q2'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:88"><i><font color="Green" title="E81">A3</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></i>;<br/>
<a NAME="E12:88"/><i><font color="Green" title="E87">A9</font></i>: 
<font color="Maroon" title="c12">o'</font>,<font color="Maroon" title="c19">q1'</font>,<font color="Maroon" title="c20">q2'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2"><b>proof </b></a><div class="add">

<a NAME="E1:88_2"/>
 <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c18">o'''</font>,<font color="Maroon" title="c12">o'</font>
 
<b>by </b><i><a class="txt" href="anproj_2.html#E6:88"><i><font color="Green" title="E83">A5</font></i></a>, <a class="txt" href="anproj_2.html#E11:88"><i><font color="Green" title="E86">A8</font></i></a>, <a class="ref" href="anproj_1.html#T35" title="ANPROJ_1:th.35">ANPROJ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:88_2"/>
<font color="Maroon" title="c12">o'</font>,<font color="Maroon" title="c19">q1'</font>,<font color="Maroon" title="c20">q2'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E11:88"><i><font color="Green" title="E86">A8</font></i></a>, <a class="ref" href="anproj_1.html#T9" title="ANPROJ_1:th.9">ANPROJ_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c21">o''''</font>, <font color="Maroon" title="c22">q1''</font>, <font color="Maroon" title="c23">q3'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E14:88"/><i><font color="Green" title="E88">A10</font></i>: 
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c21">o''''</font> &amp; <font color="Maroon" title="c6">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c22">q1''</font> &amp; <font color="Maroon" title="c8">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c23">q3'</font> &amp; <font color="Maroon" title="c21">o''''</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c22">q1''</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c23">q3'</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c21">o''''</font>,<font color="Maroon" title="c22">q1''</font>,<font color="Maroon" title="c23">q3'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E3:88"><i><font color="Green" title="E81">A3</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></i>;<br/>
<a NAME="E15:88"/><i><font color="Green" title="E89">A11</font></i>: 
<font color="Maroon" title="c12">o'</font>,<font color="Maroon" title="c19">q1'</font>,<font color="Maroon" title="c23">q3'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_3"><b>proof </b></a><div class="add">

<a NAME="E1:88_3"/>
(  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c21">o''''</font>,<font color="Maroon" title="c12">o'</font> &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c22">q1''</font>,<font color="Maroon" title="c19">q1'</font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E6:88"><i><font color="Green" title="E83">A5</font></i></a>, <a class="txt" href="anproj_2.html#E11:88"><i><font color="Green" title="E86">A8</font></i></a>, <a class="txt" href="anproj_2.html#E14:88"><i><font color="Green" title="E88">A10</font></i></a>, <a class="ref" href="anproj_1.html#T35" title="ANPROJ_1:th.35">ANPROJ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:88_3"/>
<font color="Maroon" title="c12">o'</font>,<font color="Maroon" title="c19">q1'</font>,<font color="Maroon" title="c23">q3'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E14:88"><i><font color="Green" title="E88">A10</font></i></a>, <a class="ref" href="anproj_1.html#T9" title="ANPROJ_1:th.9">ANPROJ_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E16:88"/>
not <font color="Maroon" title="c12">o'</font>,<font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c19">q1'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 
<b>by </b><i><a class="txt" href="anproj_2.html#E2:88"><i><font color="Green" title="E80">A2</font></i></a>, <a class="txt" href="anproj_2.html#E6:88"><i><font color="Green" title="E83">A5</font></i></a>, <a class="txt" href="anproj_2.html#E11:88"><i><font color="Green" title="E86">A8</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></i>;<br/>
<a NAME="E17:88"/><b>then </b><i><font color="Green" title="E90">A12</font></i>: 
<font color="Maroon" title="c12">o'</font>,<font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c14">p2'</font>,<font color="Maroon" title="c17">p3'</font>,<font color="Maroon" title="c19">q1'</font>,<font color="Maroon" title="c20">q2'</font>,<font color="Maroon" title="c23">q3'</font> <a href="anproj_2.html#R4" title="ANPROJ_2:pred.4">lie_on_an_angle</a> 
 
<b>by </b><i><a class="txt" href="anproj_2.html#E6:88"><i><font color="Green" title="E83">A5</font></i></a>, <a class="txt" href="anproj_2.html#E9:88"><i><font color="Green" title="E85">A7</font></i></a>, <a class="txt" href="anproj_2.html#E12:88"><i><font color="Green" title="E87">A9</font></i></a>, <a class="txt" href="anproj_2.html#E15:88"><i><font color="Green" title="E89">A11</font></i></a>, <a class="ref" href="anproj_2.html#D4" target="_self" title="ANPROJ_2:def.4">Def4</a></i>;<br/>
<a NAME="E18:88"/>
( not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c12">o'</font>,<font color="Maroon" title="c14">p2'</font> &amp; not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c12">o'</font>,<font color="Maroon" title="c17">p3'</font> &amp; not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c12">o'</font>,<font color="Maroon" title="c20">q2'</font> &amp; not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c12">o'</font>,<font color="Maroon" title="c23">q3'</font> &amp; not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c14">p2'</font> &amp; not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c17">p3'</font> &amp; not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c19">q1'</font>,<font color="Maroon" title="c20">q2'</font> &amp; not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c19">q1'</font>,<font color="Maroon" title="c23">q3'</font> &amp; not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c14">p2'</font>,<font color="Maroon" title="c17">p3'</font> &amp; not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c20">q2'</font>,<font color="Maroon" title="c23">q3'</font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E1:88"><i><font color="Green" title="E79">A1</font></i></a>, <a class="txt" href="anproj_2.html#E6:88"><i><font color="Green" title="E83">A5</font></i></a>, <a class="txt" href="anproj_2.html#E8:88"><i><font color="Green" title="E84">A6</font></i></a>, <a class="txt" href="anproj_2.html#E11:88"><i><font color="Green" title="E86">A8</font></i></a>, <a class="txt" href="anproj_2.html#E14:88"><i><font color="Green" title="E88">A10</font></i></a>, <a class="ref" href="anproj_1.html#T35" title="ANPROJ_1:th.35">ANPROJ_1:35</a></i>;<br/>
<a NAME="E19:88"/><b>then </b><i><font color="Green" title="E91">A13</font></i>: 
<font color="Maroon" title="c12">o'</font>,<font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c14">p2'</font>,<font color="Maroon" title="c17">p3'</font>,<font color="Maroon" title="c19">q1'</font>,<font color="Maroon" title="c20">q2'</font>,<font color="Maroon" title="c23">q3'</font> <a href="anproj_2.html#R5" title="ANPROJ_2:pred.5">are_half_mutually_not_Prop</a> 
 
<b>by </b><i><a class="ref" href="anproj_2.html#D5" target="_self" title="ANPROJ_2:def.5">Def5</a></i>;<br/>
<a NAME="E20:88"/><i><font color="Green" title="E92">A14</font></i>: 
( <font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c14">p2'</font>,<font color="Maroon" title="c17">p3'</font> <a href="anproj_2.html#R1" title="ANPROJ_2:pred.1">are_Prop_Vect</a>  &amp; <font color="Maroon" title="c19">q1'</font>,<font color="Maroon" title="c20">q2'</font>,<font color="Maroon" title="c23">q3'</font> <a href="anproj_2.html#R1" title="ANPROJ_2:pred.1">are_Prop_Vect</a>  )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E6:88"><i><font color="Green" title="E83">A5</font></i></a>, <a class="txt" href="anproj_2.html#E8:88"><i><font color="Green" title="E84">A6</font></i></a>, <a class="txt" href="anproj_2.html#E11:88"><i><font color="Green" title="E86">A8</font></i></a>, <a class="txt" href="anproj_2.html#E14:88"><i><font color="Green" title="E88">A10</font></i></a>, <a class="ref" href="anproj_2.html#D1" target="_self" title="ANPROJ_2:def.1">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c24">p1'''</font>, <font color="Maroon" title="c25">q2'''</font>, <font color="Maroon" title="c26">r3'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E22:88"/><i><font color="Green" title="E93">A15</font></i>: 
( <font color="Maroon" title="c3">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c24">p1'''</font> &amp; <font color="Maroon" title="c7">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c25">q2'''</font> &amp; <font color="Maroon" title="c11">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c26">r3'</font> &amp; <font color="Maroon" title="c24">p1'''</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c25">q2'''</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c26">r3'</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c24">p1'''</font>,<font color="Maroon" title="c25">q2'''</font>,<font color="Maroon" title="c26">r3'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E4:88"><i><font color="Green" title="E82">A4</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></i>;<br/>
<a NAME="E23:88"/><i><font color="Green" title="E94">A16</font></i>: 
<font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c20">q2'</font>,<font color="Maroon" title="c26">r3'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_4"><b>proof </b></a><div class="add">

<a NAME="E1:88_4"/>
(  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c24">p1'''</font>,<font color="Maroon" title="c13">p1'</font> &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c25">q2'''</font>,<font color="Maroon" title="c20">q2'</font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E6:88"><i><font color="Green" title="E83">A5</font></i></a>, <a class="txt" href="anproj_2.html#E11:88"><i><font color="Green" title="E86">A8</font></i></a>, <a class="txt" href="anproj_2.html#E22:88"><i><font color="Green" title="E93">A15</font></i></a>, <a class="ref" href="anproj_1.html#T35" title="ANPROJ_1:th.35">ANPROJ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:88_4"/>
<font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c20">q2'</font>,<font color="Maroon" title="c26">r3'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E22:88"><i><font color="Green" title="E93">A15</font></i></a>, <a class="ref" href="anproj_1.html#T9" title="ANPROJ_1:th.9">ANPROJ_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c27">q1'''</font>, <font color="Maroon" title="c28">p2'''</font>, <font color="Maroon" title="c29">r3''</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E25:88"/><i><font color="Green" title="E95">A17</font></i>: 
( <font color="Maroon" title="c6">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c27">q1'''</font> &amp; <font color="Maroon" title="c4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c28">p2'''</font> &amp; <font color="Maroon" title="c11">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c29">r3''</font> &amp; <font color="Maroon" title="c27">q1'''</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c28">p2'''</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c29">r3''</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c27">q1'''</font>,<font color="Maroon" title="c28">p2'''</font>,<font color="Maroon" title="c29">r3''</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E4:88"><i><font color="Green" title="E82">A4</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></i>;<br/>
<a NAME="E26:88"/><i><font color="Green" title="E96">A18</font></i>: 
<font color="Maroon" title="c19">q1'</font>,<font color="Maroon" title="c14">p2'</font>,<font color="Maroon" title="c26">r3'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_5"><b>proof </b></a><div class="add">

<a NAME="E1:88_5"/>
(  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c27">q1'''</font>,<font color="Maroon" title="c19">q1'</font> &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c28">p2'''</font>,<font color="Maroon" title="c14">p2'</font> &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c29">r3''</font>,<font color="Maroon" title="c26">r3'</font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E6:88"><i><font color="Green" title="E83">A5</font></i></a>, <a class="txt" href="anproj_2.html#E11:88"><i><font color="Green" title="E86">A8</font></i></a>, <a class="txt" href="anproj_2.html#E22:88"><i><font color="Green" title="E93">A15</font></i></a>, <a class="txt" href="anproj_2.html#E25:88"><i><font color="Green" title="E95">A17</font></i></a>, <a class="ref" href="anproj_1.html#T35" title="ANPROJ_1:th.35">ANPROJ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:88_5"/>
<font color="Maroon" title="c19">q1'</font>,<font color="Maroon" title="c14">p2'</font>,<font color="Maroon" title="c26">r3'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E25:88"><i><font color="Green" title="E95">A17</font></i></a>, <a class="ref" href="anproj_1.html#T9" title="ANPROJ_1:th.9">ANPROJ_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c30">p1''''</font>, <font color="Maroon" title="c31">q3''</font>, <font color="Maroon" title="c32">r2'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E28:88"/><i><font color="Green" title="E97">A19</font></i>: 
( <font color="Maroon" title="c3">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c30">p1''''</font> &amp; <font color="Maroon" title="c8">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c31">q3''</font> &amp; <font color="Maroon" title="c10">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c32">r2'</font> &amp; <font color="Maroon" title="c30">p1''''</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c31">q3''</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c32">r2'</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c30">p1''''</font>,<font color="Maroon" title="c31">q3''</font>,<font color="Maroon" title="c32">r2'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E4:88"><i><font color="Green" title="E82">A4</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></i>;<br/>
<a NAME="E29:88"/><i><font color="Green" title="E98">A20</font></i>: 
<font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c23">q3'</font>,<font color="Maroon" title="c32">r2'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_6"><b>proof </b></a><div class="add">

<a NAME="E1:88_6"/>
(  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c30">p1''''</font>,<font color="Maroon" title="c13">p1'</font> &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c31">q3''</font>,<font color="Maroon" title="c23">q3'</font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E6:88"><i><font color="Green" title="E83">A5</font></i></a>, <a class="txt" href="anproj_2.html#E14:88"><i><font color="Green" title="E88">A10</font></i></a>, <a class="txt" href="anproj_2.html#E28:88"><i><font color="Green" title="E97">A19</font></i></a>, <a class="ref" href="anproj_1.html#T35" title="ANPROJ_1:th.35">ANPROJ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:88_6"/>
<font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c23">q3'</font>,<font color="Maroon" title="c32">r2'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E28:88"><i><font color="Green" title="E97">A19</font></i></a>, <a class="ref" href="anproj_1.html#T9" title="ANPROJ_1:th.9">ANPROJ_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c33">p3''</font>, <font color="Maroon" title="c34">q1'''</font>, <font color="Maroon" title="c35">r2''</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E31:88"/><i><font color="Green" title="E99">A21</font></i>: 
( <font color="Maroon" title="c5">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c33">p3''</font> &amp; <font color="Maroon" title="c6">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c34">q1'''</font> &amp; <font color="Maroon" title="c10">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c35">r2''</font> &amp; <font color="Maroon" title="c33">p3''</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c34">q1'''</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c35">r2''</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c33">p3''</font>,<font color="Maroon" title="c34">q1'''</font>,<font color="Maroon" title="c35">r2''</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E4:88"><i><font color="Green" title="E82">A4</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></i>;<br/>
<a NAME="E32:88"/><i><font color="Green" title="E100">A22</font></i>: 
<font color="Maroon" title="c17">p3'</font>,<font color="Maroon" title="c19">q1'</font>,<font color="Maroon" title="c32">r2'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_7"><b>proof </b></a><div class="add">

<a NAME="E1:88_7"/>
(  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c33">p3''</font>,<font color="Maroon" title="c17">p3'</font> &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c34">q1'''</font>,<font color="Maroon" title="c19">q1'</font> &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c35">r2''</font>,<font color="Maroon" title="c32">r2'</font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E8:88"><i><font color="Green" title="E84">A6</font></i></a>, <a class="txt" href="anproj_2.html#E11:88"><i><font color="Green" title="E86">A8</font></i></a>, <a class="txt" href="anproj_2.html#E28:88"><i><font color="Green" title="E97">A19</font></i></a>, <a class="txt" href="anproj_2.html#E31:88"><i><font color="Green" title="E99">A21</font></i></a>, <a class="ref" href="anproj_1.html#T35" title="ANPROJ_1:th.35">ANPROJ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:88_7"/>
<font color="Maroon" title="c17">p3'</font>,<font color="Maroon" title="c19">q1'</font>,<font color="Maroon" title="c32">r2'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E31:88"><i><font color="Green" title="E99">A21</font></i></a>, <a class="ref" href="anproj_1.html#T9" title="ANPROJ_1:th.9">ANPROJ_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c36">p2''''</font>, <font color="Maroon" title="c37">q3'''</font>, <font color="Maroon" title="c38">r1'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E34:88"/><i><font color="Green" title="E101">A23</font></i>: 
( <font color="Maroon" title="c4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c36">p2''''</font> &amp; <font color="Maroon" title="c8">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c37">q3'''</font> &amp; <font color="Maroon" title="c9">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c38">r1'</font> &amp; <font color="Maroon" title="c36">p2''''</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c37">q3'''</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c38">r1'</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c36">p2''''</font>,<font color="Maroon" title="c37">q3'''</font>,<font color="Maroon" title="c38">r1'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E4:88"><i><font color="Green" title="E82">A4</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></i>;<br/>
<a NAME="E35:88"/><i><font color="Green" title="E102">A24</font></i>: 
<font color="Maroon" title="c14">p2'</font>,<font color="Maroon" title="c23">q3'</font>,<font color="Maroon" title="c38">r1'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_8"><b>proof </b></a><div class="add">

<a NAME="E1:88_8"/>
(  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c36">p2''''</font>,<font color="Maroon" title="c14">p2'</font> &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c37">q3'''</font>,<font color="Maroon" title="c23">q3'</font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E6:88"><i><font color="Green" title="E83">A5</font></i></a>, <a class="txt" href="anproj_2.html#E14:88"><i><font color="Green" title="E88">A10</font></i></a>, <a class="txt" href="anproj_2.html#E34:88"><i><font color="Green" title="E101">A23</font></i></a>, <a class="ref" href="anproj_1.html#T35" title="ANPROJ_1:th.35">ANPROJ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:88_8"/>
<font color="Maroon" title="c14">p2'</font>,<font color="Maroon" title="c23">q3'</font>,<font color="Maroon" title="c38">r1'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E34:88"><i><font color="Green" title="E101">A23</font></i></a>, <a class="ref" href="anproj_1.html#T9" title="ANPROJ_1:th.9">ANPROJ_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c39">p3'''</font>, <font color="Maroon" title="c40">q2''''</font>, <font color="Maroon" title="c41">r1''</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E37:88"/><i><font color="Green" title="E103">A25</font></i>: 
( <font color="Maroon" title="c5">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c39">p3'''</font> &amp; <font color="Maroon" title="c7">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c40">q2''''</font> &amp; <font color="Maroon" title="c9">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K3" title="ANPROJ_1:func.3">Dir</a> <font color="Maroon" title="c41">r1''</font> &amp; <font color="Maroon" title="c39">p3'''</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c40">q2''''</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c41">r1''</font> <a href="anproj_1.html#NR1" title="ANPROJ_1:NR.1">is_Prop_Vect</a>  &amp; <font color="Maroon" title="c39">p3'''</font>,<font color="Maroon" title="c40">q2''''</font>,<font color="Maroon" title="c41">r1''</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="anproj_2.html#E4:88"><i><font color="Green" title="E82">A4</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></i>;<br/>
<a NAME="E38:88"/><i><font color="Green" title="E104">A26</font></i>: 
<font color="Maroon" title="c17">p3'</font>,<font color="Maroon" title="c20">q2'</font>,<font color="Maroon" title="c38">r1'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_9"><b>proof </b></a><div class="add">

<a NAME="E1:88_9"/>
(  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c39">p3'''</font>,<font color="Maroon" title="c17">p3'</font> &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c40">q2''''</font>,<font color="Maroon" title="c20">q2'</font> &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c41">r1''</font>,<font color="Maroon" title="c38">r1'</font> )
 
<b>by </b><i><a class="txt" href="anproj_2.html#E8:88"><i><font color="Green" title="E84">A6</font></i></a>, <a class="txt" href="anproj_2.html#E11:88"><i><font color="Green" title="E86">A8</font></i></a>, <a class="txt" href="anproj_2.html#E34:88"><i><font color="Green" title="E101">A23</font></i></a>, <a class="txt" href="anproj_2.html#E37:88"><i><font color="Green" title="E103">A25</font></i></a>, <a class="ref" href="anproj_1.html#T35" title="ANPROJ_1:th.35">ANPROJ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E2:88_9"/>
<font color="Maroon" title="c17">p3'</font>,<font color="Maroon" title="c20">q2'</font>,<font color="Maroon" title="c38">r1'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E37:88"><i><font color="Green" title="E103">A25</font></i></a>, <a class="ref" href="anproj_1.html#T9" title="ANPROJ_1:th.9">ANPROJ_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E39:88"/>
<font color="Maroon" title="c38">r1'</font>,<font color="Maroon" title="c32">r2'</font>,<font color="Maroon" title="c26">r3'</font> <a href="anproj_2.html#R1" title="ANPROJ_2:pred.1">are_Prop_Vect</a> 
 
<b>by </b><i><a class="txt" href="anproj_2.html#E22:88"><i><font color="Green" title="E93">A15</font></i></a>, <a class="txt" href="anproj_2.html#E28:88"><i><font color="Green" title="E97">A19</font></i></a>, <a class="txt" href="anproj_2.html#E34:88"><i><font color="Green" title="E101">A23</font></i></a>, <a class="ref" href="anproj_2.html#D1" target="_self" title="ANPROJ_2:def.1">Def1</a></i>;<br/>
<a NAME="E40:88"/><b>then </b>
<font color="Maroon" title="c38">r1'</font>,<font color="Maroon" title="c32">r2'</font>,<font color="Maroon" title="c26">r3'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 
<b>by </b><i><a class="txt" href="anproj_2.html#E6:88"><i><font color="Green" title="E83">A5</font></i></a>, <a class="txt" href="anproj_2.html#E17:88"><i><font color="Green" title="E90">A12</font></i></a>, <a class="txt" href="anproj_2.html#E19:88"><i><font color="Green" title="E91">A13</font></i></a>, <a class="txt" href="anproj_2.html#E20:88"><i><font color="Green" title="E92">A14</font></i></a>, <a class="txt" href="anproj_2.html#E23:88"><i><font color="Green" title="E94">A16</font></i></a>, <a class="txt" href="anproj_2.html#E26:88"><i><font color="Green" title="E96">A18</font></i></a>, <a class="txt" href="anproj_2.html#E29:88"><i><font color="Green" title="E98">A20</font></i></a>, <a class="txt" href="anproj_2.html#E32:88"><i><font color="Green" title="E100">A22</font></i></a>, <a class="txt" href="anproj_2.html#E35:88"><i><font color="Green" title="E102">A24</font></i></a>, <a class="txt" href="anproj_2.html#E38:88"><i><font color="Green" title="E104">A26</font></i></a>, <a class="ref" href="anproj_2.html#T9" target="_self" title="ANPROJ_2:th.9">Th9</a></i>;<br/>
<b>hence </b><a NAME="E41:88"/>
<font color="Maroon" title="c9">r1</font>,<font color="Maroon" title="c10">r2</font>,<font color="Maroon" title="c11">r3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <b>by </b><i><a class="txt" href="anproj_2.html#E22:88"><i><font color="Green" title="E93">A15</font></i></a>, <a class="txt" href="anproj_2.html#E28:88"><i><font color="Green" title="E97">A19</font></i></a>, <a class="txt" href="anproj_2.html#E34:88"><i><font color="Green" title="E101">A23</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
