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<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#V5" title="ANPROJ_2:attr.5">Desarguesian</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#D12" target="_self" title="ANPROJ_2:def.12">ANPROJ_2:def 12</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="c6">q1</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="c6">q1</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="c4">p2</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="c5">p3</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="c4">p2</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; not <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; not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">p2</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="c3">p1</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="c6">q1</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="c4">p2</font>,<font color="Maroon" title="c5">p3</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="c7">q2</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="c3">p1</font>,<font color="Maroon" title="c5">p3</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="c6">q1</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="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="c4">p2</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="c5">p3</font>,<font color="Maroon" title="c8">q3</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="E1:86"/><i><font color="Green" title="E76">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="c6">q1</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="c6">q1</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="c4">p2</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="c5">p3</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="E2:86"/><i><font color="Green" title="E77">A2</font></i>: 
( not <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; not <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; not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c5">p3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>and </b><br/><a NAME="E3:86"/><i><font color="Green" title="E78">A3</font></i>: 
( <font color="Maroon" title="c3">p1</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="c6">q1</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="c4">p2</font>,<font color="Maroon" title="c5">p3</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="c7">q2</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="c3">p1</font>,<font color="Maroon" title="c5">p3</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="c6">q1</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>  )
 <b>and </b><br/><a NAME="E4:86"/><i><font color="Green" title="E79">A4</font></i>: 
( <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="c4">p2</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="c5">p3</font>,<font color="Maroon" title="c8">q3</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> 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="E6:86"/><i><font color="Green" title="E80">A5</font></i>: 
( not <font color="Maroon" title="c12">o'</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; <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> )
 <b>by </b><i><a class="ref" href="anproj_1.html#T42" title="ANPROJ_1:th.42">ANPROJ_1:42</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">p1'</font>, <font color="Maroon" title="c14">p2'</font>, <font color="Maroon" title="c15">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="E8:86"/><i><font color="Green" title="E81">A6</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="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="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="c15">r3'</font> &amp; not <font color="Maroon" title="c13">p1'</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c14">p2'</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c15">r3'</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; <font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c14">p2'</font>,<font color="Maroon" title="c15">r3'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="#E3:86"><i><font color="Green" title="E78">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="c16">q1'</font>, <font color="Maroon" title="c17">q2'</font>, <font color="Maroon" title="c18">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="E10:86"/><i><font color="Green" title="E82">A7</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="c16">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="c17">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="c18">r3''</font> &amp; not <font color="Maroon" title="c16">q1'</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c17">q2'</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c18">r3''</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; <font color="Maroon" title="c16">q1'</font>,<font color="Maroon" title="c17">q2'</font>,<font color="Maroon" title="c18">r3''</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="#E3:86"><i><font color="Green" title="E78">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="E11:86"/><i><font color="Green" title="E83">A8</font></i>: 
<font color="Maroon" title="c16">q1'</font>,<font color="Maroon" title="c17">q2'</font>,<font color="Maroon" title="c15">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="86_1"><b>proof </b></a><div class="add">

<a NAME="E1:86_1"/>
 <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c18">r3''</font>,<font color="Maroon" title="c15">r3'</font>
 
<b>by </b><i><a class="txt" href="#E8:86"><i><font color="Green" title="E81">A6</font></i></a>, <a class="txt" href="#E10:86"><i><font color="Green" title="E82">A7</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="E2:86_1"/>
<font color="Maroon" title="c16">q1'</font>,<font color="Maroon" title="c17">q2'</font>,<font color="Maroon" title="c15">r3'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="#E10:86"><i><font color="Green" title="E82">A7</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="c19">p2''</font>, <font color="Maroon" title="c20">p3'</font>, <font color="Maroon" title="c21">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="E13:86"/><i><font color="Green" title="E84">A9</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="c19">p2''</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="c20">p3'</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="c21">r1'</font> &amp; not <font color="Maroon" title="c19">p2''</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c20">p3'</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c21">r1'</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; <font color="Maroon" title="c19">p2''</font>,<font color="Maroon" title="c20">p3'</font>,<font color="Maroon" title="c21">r1'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="#E3:86"><i><font color="Green" title="E78">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="E14:86"/><i><font color="Green" title="E85">A10</font></i>: 
<font color="Maroon" title="c14">p2'</font>,<font color="Maroon" title="c20">p3'</font>,<font color="Maroon" title="c21">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="86_2"><b>proof </b></a><div class="add">

<a NAME="E1:86_2"/>
 <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c19">p2''</font>,<font color="Maroon" title="c14">p2'</font>
 
<b>by </b><i><a class="txt" href="#E8:86"><i><font color="Green" title="E81">A6</font></i></a>, <a class="txt" href="#E13:86"><i><font color="Green" title="E84">A9</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="E2:86_2"/>
<font color="Maroon" title="c14">p2'</font>,<font color="Maroon" title="c20">p3'</font>,<font color="Maroon" title="c21">r1'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="#E13:86"><i><font color="Green" title="E84">A9</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="c22">q2''</font>, <font color="Maroon" title="c23">q3'</font>, <font color="Maroon" title="c24">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="E16:86"/><i><font color="Green" title="E86">A11</font></i>: 
( <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="c22">q2''</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="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="c24">r1''</font> &amp; not <font color="Maroon" title="c22">q2''</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c23">q3'</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c24">r1''</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; <font color="Maroon" title="c22">q2''</font>,<font color="Maroon" title="c23">q3'</font>,<font color="Maroon" title="c24">r1''</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="#E3:86"><i><font color="Green" title="E78">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="E17:86"/><i><font color="Green" title="E87">A12</font></i>: 
<font color="Maroon" title="c17">q2'</font>,<font color="Maroon" title="c23">q3'</font>,<font color="Maroon" title="c21">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="86_3"><b>proof </b></a><div class="add">

<a NAME="E1:86_3"/>
(  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c22">q2''</font>,<font color="Maroon" title="c17">q2'</font> &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c24">r1''</font>,<font color="Maroon" title="c21">r1'</font> )
 
<b>by </b><i><a class="txt" href="#E10:86"><i><font color="Green" title="E82">A7</font></i></a>, <a class="txt" href="#E13:86"><i><font color="Green" title="E84">A9</font></i></a>, <a class="txt" href="#E16:86"><i><font color="Green" title="E86">A11</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="E2:86_3"/>
<font color="Maroon" title="c17">q2'</font>,<font color="Maroon" title="c23">q3'</font>,<font color="Maroon" title="c21">r1'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="#E16:86"><i><font color="Green" title="E86">A11</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="c25">p1''</font>, <font color="Maroon" title="c26">p3''</font>, <font color="Maroon" title="c27">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="E19:86"/><i><font color="Green" title="E88">A13</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="c25">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="c26">p3''</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="c27">r2'</font> &amp; not <font color="Maroon" title="c25">p1''</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c26">p3''</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c27">r2'</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; <font color="Maroon" title="c25">p1''</font>,<font color="Maroon" title="c26">p3''</font>,<font color="Maroon" title="c27">r2'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="#E3:86"><i><font color="Green" title="E78">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="E20:86"/><i><font color="Green" title="E89">A14</font></i>: 
<font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c20">p3'</font>,<font color="Maroon" title="c27">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="86_4"><b>proof </b></a><div class="add">

<a NAME="E1:86_4"/>
(  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c25">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="c26">p3''</font>,<font color="Maroon" title="c20">p3'</font> )
 
<b>by </b><i><a class="txt" href="#E8:86"><i><font color="Green" title="E81">A6</font></i></a>, <a class="txt" href="#E13:86"><i><font color="Green" title="E84">A9</font></i></a>, <a class="txt" href="#E19:86"><i><font color="Green" title="E88">A13</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="E2:86_4"/>
<font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c20">p3'</font>,<font color="Maroon" title="c27">r2'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="#E19:86"><i><font color="Green" title="E88">A13</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="c28">q1''</font>, <font color="Maroon" title="c29">q3''</font>, <font color="Maroon" title="c30">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="E22:86"/><i><font color="Green" title="E90">A15</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="c28">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="c29">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="c30">r2''</font> &amp; not <font color="Maroon" title="c28">q1''</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c29">q3''</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c30">r2''</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; <font color="Maroon" title="c28">q1''</font>,<font color="Maroon" title="c29">q3''</font>,<font color="Maroon" title="c30">r2''</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="#E3:86"><i><font color="Green" title="E78">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="E23:86"/><i><font color="Green" title="E91">A16</font></i>: 
<font color="Maroon" title="c16">q1'</font>,<font color="Maroon" title="c23">q3'</font>,<font color="Maroon" title="c27">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="86_5"><b>proof </b></a><div class="add">

<a NAME="E1:86_5"/>
(  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c28">q1''</font>,<font color="Maroon" title="c16">q1'</font> &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c29">q3''</font>,<font color="Maroon" title="c23">q3'</font> &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c30">r2''</font>,<font color="Maroon" title="c27">r2'</font> )
 
<b>by </b><i><a class="txt" href="#E10:86"><i><font color="Green" title="E82">A7</font></i></a>, <a class="txt" href="#E16:86"><i><font color="Green" title="E86">A11</font></i></a>, <a class="txt" href="#E19:86"><i><font color="Green" title="E88">A13</font></i></a>, <a class="txt" href="#E22:86"><i><font color="Green" title="E90">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="E2:86_5"/>
<font color="Maroon" title="c16">q1'</font>,<font color="Maroon" title="c23">q3'</font>,<font color="Maroon" title="c27">r2'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="#E22:86"><i><font color="Green" title="E90">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>
<a NAME="E24:86"/><i><font color="Green" title="E92">A17</font></i>: 
( 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="c16">q1'</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">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="c16">q1'</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">q2'</font> &amp; not  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c20">p3'</font>,<font color="Maroon" title="c23">q3'</font> )
 
<b>by </b><i><a class="txt" href="#E1:86"><i><font color="Green" title="E76">A1</font></i></a>, <a class="txt" href="#E6:86"><i><font color="Green" title="E80">A5</font></i></a>, <a class="txt" href="#E8:86"><i><font color="Green" title="E81">A6</font></i></a>, <a class="txt" href="#E10:86"><i><font color="Green" title="E82">A7</font></i></a>, <a class="txt" href="#E13:86"><i><font color="Green" title="E84">A9</font></i></a>, <a class="txt" href="#E16:86"><i><font color="Green" title="E86">A11</font></i></a>, <a class="ref" href="anproj_1.html#T35" title="ANPROJ_1:th.35">ANPROJ_1:35</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c31">o''</font>, <font color="Maroon" title="c32">p1'''</font>, <font color="Maroon" title="c33">q1'''</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="E26:86"/><i><font color="Green" title="E93">A18</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="c31">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="c32">p1'''</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="c33">q1'''</font> &amp; not <font color="Maroon" title="c31">o''</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c32">p1'''</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c33">q1'''</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; <font color="Maroon" title="c31">o''</font>,<font color="Maroon" title="c32">p1'''</font>,<font color="Maroon" title="c33">q1'''</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="#E4:86"><i><font color="Green" title="E79">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="E27:86"/><i><font color="Green" title="E94">A19</font></i>: 
<font color="Maroon" title="c12">o'</font>,<font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c16">q1'</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="86_6"><b>proof </b></a><div class="add">

<a NAME="E1:86_6"/>
(  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c31">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="c32">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="c33">q1'''</font>,<font color="Maroon" title="c16">q1'</font> )
 
<b>by </b><i><a class="txt" href="#E6:86"><i><font color="Green" title="E80">A5</font></i></a>, <a class="txt" href="#E8:86"><i><font color="Green" title="E81">A6</font></i></a>, <a class="txt" href="#E10:86"><i><font color="Green" title="E82">A7</font></i></a>, <a class="txt" href="#E26:86"><i><font color="Green" title="E93">A18</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="E2:86_6"/>
<font color="Maroon" title="c12">o'</font>,<font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c16">q1'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="#E26:86"><i><font color="Green" title="E93">A18</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="c34">o'''</font>, <font color="Maroon" title="c35">p2'''</font>, <font color="Maroon" title="c36">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="E29:86"/><i><font color="Green" title="E95">A20</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="c34">o'''</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="c35">p2'''</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="c36">q2'''</font> &amp; not <font color="Maroon" title="c34">o'''</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c35">p2'''</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c36">q2'''</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; <font color="Maroon" title="c34">o'''</font>,<font color="Maroon" title="c35">p2'''</font>,<font color="Maroon" title="c36">q2'''</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="#E4:86"><i><font color="Green" title="E79">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="E30:86"/><i><font color="Green" title="E96">A21</font></i>: 
<font color="Maroon" title="c12">o'</font>,<font color="Maroon" title="c14">p2'</font>,<font color="Maroon" title="c17">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="86_7"><b>proof </b></a><div class="add">

<a NAME="E1:86_7"/>
(  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c34">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="c35">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="c36">q2'''</font>,<font color="Maroon" title="c17">q2'</font> )
 
<b>by </b><i><a class="txt" href="#E6:86"><i><font color="Green" title="E80">A5</font></i></a>, <a class="txt" href="#E8:86"><i><font color="Green" title="E81">A6</font></i></a>, <a class="txt" href="#E10:86"><i><font color="Green" title="E82">A7</font></i></a>, <a class="txt" href="#E29:86"><i><font color="Green" title="E95">A20</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="E2:86_7"/>
<font color="Maroon" title="c12">o'</font>,<font color="Maroon" title="c14">p2'</font>,<font color="Maroon" title="c17">q2'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 <b>by </b><i><a class="txt" href="#E29:86"><i><font color="Green" title="E95">A20</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="c37">o''''</font>, <font color="Maroon" title="c38">p3'''</font>, <font color="Maroon" title="c39">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="E32:86"/><i><font color="Green" title="E97">A22</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="c37">o''''</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="c38">p3'''</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="c39">q3'''</font> &amp; not <font color="Maroon" title="c37">o''''</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c38">p3'''</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; not <font color="Maroon" title="c39">q3'''</font> is <a href="struct_0.html#V12" title="STRUCT_0:attr.12">zero</a> &amp; <font color="Maroon" title="c37">o''''</font>,<font color="Maroon" title="c38">p3'''</font>,<font color="Maroon" title="c39">q3'''</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a>  )
 <b>by </b><i><a class="txt" href="#E4:86"><i><font color="Green" title="E79">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="E33:86"/><i><font color="Green" title="E98">A23</font></i>: 
<font color="Maroon" title="c12">o'</font>,<font color="Maroon" title="c20">p3'</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="86_8"><b>proof </b></a><div class="add">

<a NAME="E1:86_8"/>
(  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c37">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="c38">p3'''</font>,<font color="Maroon" title="c20">p3'</font> &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c39">q3'''</font>,<font color="Maroon" title="c23">q3'</font> )
 
<b>by </b><i><a class="txt" href="#E6:86"><i><font color="Green" title="E80">A5</font></i></a>, <a class="txt" href="#E13:86"><i><font color="Green" title="E84">A9</font></i></a>, <a class="txt" href="#E16:86"><i><font color="Green" title="E86">A11</font></i></a>, <a class="txt" href="#E32:86"><i><font color="Green" title="E97">A22</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="E2:86_8"/>
<font color="Maroon" title="c12">o'</font>,<font color="Maroon" title="c20">p3'</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="#E32:86"><i><font color="Green" title="E97">A22</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="E34:86"/><i><font color="Green" title="E99">A24</font></i>: 
<font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c14">p2'</font>,<font color="Maroon" title="c20">p3'</font> <a href="anproj_2.html#R1" title="ANPROJ_2:pred.1">are_Prop_Vect</a> 
 
<b>by </b><i><a class="txt" href="#E8:86"><i><font color="Green" title="E81">A6</font></i></a>, <a class="txt" href="#E13:86"><i><font color="Green" title="E84">A9</font></i></a>, <a class="ref" href="anproj_2.html#D1" target="_self" title="ANPROJ_2:def.1">Def1</a></i>;<br/>
<a NAME="E35:86"/><i><font color="Green" title="E100">A25</font></i>: 
<font color="Maroon" title="c16">q1'</font>,<font color="Maroon" title="c17">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="#E10:86"><i><font color="Green" title="E82">A7</font></i></a>, <a class="txt" href="#E16:86"><i><font color="Green" title="E86">A11</font></i></a>, <a class="ref" href="anproj_2.html#D1" target="_self" title="ANPROJ_2:def.1">Def1</a></i>;<br/>
<a NAME="E36:86"/><i><font color="Green" title="E101">A26</font></i>: 
<font color="Maroon" title="c21">r1'</font>,<font color="Maroon" title="c27">r2'</font>,<font color="Maroon" title="c15">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="#E8:86"><i><font color="Green" title="E81">A6</font></i></a>, <a class="txt" href="#E13:86"><i><font color="Green" title="E84">A9</font></i></a>, <a class="txt" href="#E19:86"><i><font color="Green" title="E88">A13</font></i></a>, <a class="ref" href="anproj_2.html#D1" target="_self" title="ANPROJ_2:def.1">Def1</a></i>;<br/>
<a NAME="E37:86"/><i><font color="Green" title="E102">A27</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="c20">p3'</font>,<font color="Maroon" title="c16">q1'</font>,<font color="Maroon" title="c17">q2'</font>,<font color="Maroon" title="c23">q3'</font> <a href="anproj_2.html#R3" title="ANPROJ_2:pred.3">are_perspective</a> 
 
<b>by </b><i><a class="txt" href="#E27:86"><i><font color="Green" title="E94">A19</font></i></a>, <a class="txt" href="#E30:86"><i><font color="Green" title="E96">A21</font></i></a>, <a class="txt" href="#E33:86"><i><font color="Green" title="E98">A23</font></i></a>, <a class="ref" href="anproj_2.html#D3" target="_self" title="ANPROJ_2:def.3">Def3</a></i>;<br/>
<a NAME="E38:86"/><i><font color="Green" title="E103">A28</font></i>: 
not <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="#E2:86"><i><font color="Green" title="E77">A2</font></i></a>, <a class="txt" href="#E6:86"><i><font color="Green" title="E80">A5</font></i></a>, <a class="txt" href="#E8:86"><i><font color="Green" title="E81">A6</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></i>;<br/>
<a NAME="E39:86"/><i><font color="Green" title="E104">A29</font></i>: 
not <font color="Maroon" title="c12">o'</font>,<font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c20">p3'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 
<b>by </b><i><a class="txt" href="#E2:86"><i><font color="Green" title="E77">A2</font></i></a>, <a class="txt" href="#E6:86"><i><font color="Green" title="E80">A5</font></i></a>, <a class="txt" href="#E8:86"><i><font color="Green" title="E81">A6</font></i></a>, <a class="txt" href="#E13:86"><i><font color="Green" title="E84">A9</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></i>;<br/>
<a NAME="E40:86"/><i><font color="Green" title="E105">A30</font></i>: 
not <font color="Maroon" title="c12">o'</font>,<font color="Maroon" title="c14">p2'</font>,<font color="Maroon" title="c20">p3'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 
<b>by </b><i><a class="txt" href="#E2:86"><i><font color="Green" title="E77">A2</font></i></a>, <a class="txt" href="#E6:86"><i><font color="Green" title="E80">A5</font></i></a>, <a class="txt" href="#E8:86"><i><font color="Green" title="E81">A6</font></i></a>, <a class="txt" href="#E13:86"><i><font color="Green" title="E84">A9</font></i></a>, <a class="ref" href="anproj_2.html#T24" target="_self" title="ANPROJ_2:th.24">Th24</a></i>;<br/>
<a NAME="E41:86"/><i><font color="Green" title="E106">A31</font></i>: 
<font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c14">p2'</font>,<font color="Maroon" title="c20">p3'</font>,<font color="Maroon" title="c21">r1'</font>,<font color="Maroon" title="c27">r2'</font>,<font color="Maroon" title="c15">r3'</font> <a href="anproj_2.html#R2" title="ANPROJ_2:pred.2">lie_on_a_triangle</a> 
 
<b>by </b><i><a class="txt" href="#E8:86"><i><font color="Green" title="E81">A6</font></i></a>, <a class="txt" href="#E14:86"><i><font color="Green" title="E85">A10</font></i></a>, <a class="txt" href="#E20:86"><i><font color="Green" title="E89">A14</font></i></a>, <a class="ref" href="anproj_2.html#D2" target="_self" title="ANPROJ_2:def.2">Def2</a></i>;<br/>
<a NAME="E42:86"/>
<font color="Maroon" title="c16">q1'</font>,<font color="Maroon" title="c17">q2'</font>,<font color="Maroon" title="c23">q3'</font>,<font color="Maroon" title="c21">r1'</font>,<font color="Maroon" title="c27">r2'</font>,<font color="Maroon" title="c15">r3'</font> <a href="anproj_2.html#R2" title="ANPROJ_2:pred.2">lie_on_a_triangle</a> 
 
<b>by </b><i><a class="txt" href="#E11:86"><i><font color="Green" title="E83">A8</font></i></a>, <a class="txt" href="#E17:86"><i><font color="Green" title="E87">A12</font></i></a>, <a class="txt" href="#E23:86"><i><font color="Green" title="E91">A16</font></i></a>, <a class="ref" href="anproj_2.html#D2" target="_self" title="ANPROJ_2:def.2">Def2</a></i>;<br/>
<a NAME="E43:86"/><b>then </b>
<font color="Maroon" title="c21">r1'</font>,<font color="Maroon" title="c27">r2'</font>,<font color="Maroon" title="c15">r3'</font> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 
<b>by </b><i><a class="txt" href="#E6:86"><i><font color="Green" title="E80">A5</font></i></a>, <a class="txt" href="#E24:86"><i><font color="Green" title="E92">A17</font></i></a>, <a class="txt" href="#E34:86"><i><font color="Green" title="E99">A24</font></i></a>, <a class="txt" href="#E35:86"><i><font color="Green" title="E100">A25</font></i></a>, <a class="txt" href="#E36:86"><i><font color="Green" title="E101">A26</font></i></a>, <a class="txt" href="#E37:86"><i><font color="Green" title="E102">A27</font></i></a>, <a class="txt" href="#E38:86"><i><font color="Green" title="E103">A28</font></i></a>, <a class="txt" href="#E39:86"><i><font color="Green" title="E104">A29</font></i></a>, <a class="txt" href="#E40:86"><i><font color="Green" title="E105">A30</font></i></a>, <a class="txt" href="#E41:86"><i><font color="Green" title="E106">A31</font></i></a>, <a class="ref" href="anproj_2.html#T8" target="_self" title="ANPROJ_2:th.8">Th8</a></i>;<br/>
<b>hence </b><a NAME="E44:86"/>
<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="#E8:86"><i><font color="Green" title="E81">A6</font></i></a>, <a class="txt" href="#E13:86"><i><font color="Green" title="E84">A9</font></i></a>, <a class="txt" href="#E19:86"><i><font color="Green" title="E88">A13</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>
