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<span class="kw">let </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K5" title="ANPROJ_1:func.5">ProjectiveSpace</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 3</span>)</span></span>)</span>  st <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> &amp; ( <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c6">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) holds <br/>( (  not  <a href="anproj_9.html#K5" title="ANPROJ_9:func.5">Dir100</a> , <a href="anproj_9.html#K6" title="ANPROJ_9:func.6">Dir010</a> ,<font color="Olive" title="b1">P</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a>  implies <font color="Maroon" title="c6">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) &amp; (  not  <a href="anproj_9.html#K5" title="ANPROJ_9:func.5">Dir100</a> , <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</a> ,<font color="Olive" title="b1">P</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a>  implies <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) &amp; (  not  <a href="anproj_9.html#K6" title="ANPROJ_9:func.6">Dir010</a> , <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</a> ,<font color="Olive" title="b1">P</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a>  implies <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">P</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K5" title="ANPROJ_1:func.5">ProjectiveSpace</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 3</span>)</span></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> &amp; ( <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c6">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) implies ( (  not  <a href="anproj_9.html#K5" title="ANPROJ_9:func.5">Dir100</a> , <a href="anproj_9.html#K6" title="ANPROJ_9:func.6">Dir010</a> ,<font color="Maroon" title="c7">P</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a>  implies <font color="Maroon" title="c6">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) &amp; (  not  <a href="anproj_9.html#K5" title="ANPROJ_9:func.5">Dir100</a> , <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</a> ,<font color="Maroon" title="c7">P</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a>  implies <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) &amp; (  not  <a href="anproj_9.html#K6" title="ANPROJ_9:func.6">Dir010</a> , <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</a> ,<font color="Maroon" title="c7">P</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a>  implies <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:45"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>
 <span class="kw">and </span><br/><a NAME="E2:45"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c6">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  not  <a href="anproj_9.html#K5" title="ANPROJ_9:func.5">Dir100</a> , <a href="anproj_9.html#K6" title="ANPROJ_9:func.6">Dir010</a> ,<font color="Maroon" title="c7">P</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a>  implies <font color="Maroon" title="c6">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) &amp; (  not  <a href="anproj_9.html#K5" title="ANPROJ_9:func.5">Dir100</a> , <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</a> ,<font color="Maroon" title="c7">P</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a>  implies <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) &amp; (  not  <a href="anproj_9.html#K6" title="ANPROJ_9:func.6">Dir010</a> , <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</a> ,<font color="Maroon" title="c7">P</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a>  implies <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E3:45"/>
(  not  <a href="anproj_9.html#K5" title="ANPROJ_9:func.5">Dir100</a> , <a href="anproj_9.html#K6" title="ANPROJ_9:func.6">Dir010</a> ,<font color="Maroon" title="c7">P</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a>  implies <font color="Maroon" title="c6">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  not  <a href="anproj_9.html#K5" title="ANPROJ_9:func.5">Dir100</a> , <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</a> ,<font color="Maroon" title="c7">P</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a>  implies <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) &amp; (  not  <a href="anproj_9.html#K6" title="ANPROJ_9:func.6">Dir010</a> , <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</a> ,<font color="Maroon" title="c7">P</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a>  implies <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:45_1"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
 not  <a href="anproj_9.html#K5" title="ANPROJ_9:func.5">Dir100</a> , <a href="anproj_9.html#K6" title="ANPROJ_9:func.6">Dir010</a> ,<font color="Maroon" title="c7">P</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/>

<span class="kw">assume </span><a NAME="E2:45_1"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:45_1"/><span class="lab"><font color="Green" title="E37">A5</font></span>: 
 not <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="euclid_5.html#T4" onmouseover="rs('euclid_5/T4')" onmouseout="rh()">EUCLID_5:4</a>, <a class="ref" href="finseq_1.html#T78" onmouseover="rs('finseq_1/T78')" onmouseout="rh()">FINSEQ_1:78</a></span>;<br/>
<a NAME="E4:45_1"/><span class="lab"><font color="Green" title="E38">A6</font></span>: 
 not <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default">1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>,<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>,<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="anproj_2.html#T23" onmouseover="rs('anproj_2/T23')" onmouseout="rh()">ANPROJ_2:23</a>, <a class="ref" href="anproj_9.html#T10" target="_self" onmouseover="rs('anproj_9/T10')" onmouseout="rh()">Th11</a>, <a class="txt" href="#E3:45_1"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E33">A1</font></span></a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default">1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T8" onmouseover="rs('euclid_5/T8')" onmouseout="rh()">EUCLID_5:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">a</font></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T8" onmouseover="rs('euclid_5/T8')" onmouseout="rh()">EUCLID_5:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">a</font></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">b</font></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">a</font></span>)</span>,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">b</font></span>)</span>,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">c</font></span>)</span></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T8" onmouseover="rs('euclid_5/T8')" onmouseout="rh()">EUCLID_5:8</a></span>
<br/>.= 
<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">b</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T6" onmouseover="rs('euclid_5/T6')" onmouseout="rh()">EUCLID_5:6</a></span>
<br/>.= 
<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">b</font></span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">c</font></span>)</span></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T6" onmouseover="rs('euclid_5/T6')" onmouseout="rh()">EUCLID_5:6</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 3</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="euclid_5.html#T4" onmouseover="rs('euclid_5/T4')" onmouseout="rh()">EUCLID_5:4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:45_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_1"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="ref" href="anproj_1.html#D2" onmouseover="rs('anproj_1/D2')" onmouseout="rh()">ANPROJ_1:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E4:45"/>
(  not  <a href="anproj_9.html#K5" title="ANPROJ_9:func.5">Dir100</a> , <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</a> ,<font color="Maroon" title="c7">P</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a>  implies <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  <a href="anproj_9.html#K6" title="ANPROJ_9:func.6">Dir010</a> , <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</a> ,<font color="Maroon" title="c7">P</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a>  implies <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:45_2"/><span class="lab"><font color="Green" title="E35">A7</font></span>: 
 not  <a href="anproj_9.html#K5" title="ANPROJ_9:func.5">Dir100</a> , <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</a> ,<font color="Maroon" title="c7">P</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/>

<span class="kw">assume </span><a NAME="E2:45_2"/><span class="lab"><font color="Green" title="E36">A8</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:45_2"/><span class="lab"><font color="Green" title="E37">A9</font></span>: 
 not <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="euclid_5.html#T4" onmouseover="rs('euclid_5/T4')" onmouseout="rh()">EUCLID_5:4</a>, <a class="ref" href="finseq_1.html#T78" onmouseover="rs('finseq_1/T78')" onmouseout="rh()">FINSEQ_1:78</a></span>;<br/>
<a NAME="E4:45_2"/><span class="lab"><font color="Green" title="E38">A10</font></span>: 
 not <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default">1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>,<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>,<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_2"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="ref" href="anproj_2.html#T23" onmouseover="rs('anproj_2/T23')" onmouseout="rh()">ANPROJ_2:23</a>, <a class="ref" href="anproj_9.html#T10" target="_self" onmouseover="rs('anproj_9/T10')" onmouseout="rh()">Th11</a>, <a class="txt" href="#E3:45_2"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E33">A1</font></span></a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default">1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T8" onmouseover="rs('euclid_5/T8')" onmouseout="rh()">EUCLID_5:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">a</font></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T8" onmouseover="rs('euclid_5/T8')" onmouseout="rh()">EUCLID_5:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">a</font></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">c</font></span>)</span></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">a</font></span>)</span>,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">b</font></span>)</span>,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">c</font></span>)</span></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T8" onmouseover="rs('euclid_5/T8')" onmouseout="rh()">EUCLID_5:8</a></span>
<br/>.= 
<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">c</font></span>)</span></span>)</span></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T6" onmouseover="rs('euclid_5/T6')" onmouseout="rh()">EUCLID_5:6</a></span>
<br/>.= 
<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">b</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">c</font></span>)</span></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T6" onmouseover="rs('euclid_5/T6')" onmouseout="rh()">EUCLID_5:6</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 3</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_2"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="ref" href="euclid_5.html#T4" onmouseover="rs('euclid_5/T4')" onmouseout="rh()">EUCLID_5:4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:45_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_2"><span class="lab"><font color="Green" title="E38">A10</font></span></a>, <a class="ref" href="anproj_1.html#D2" onmouseover="rs('anproj_1/D2')" onmouseout="rh()">ANPROJ_1:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E5:45"/>
(  not  <a href="anproj_9.html#K6" title="ANPROJ_9:func.6">Dir010</a> , <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</a> ,<font color="Maroon" title="c7">P</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a>  implies <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:45_3"/><span class="lab"><font color="Green" title="E35">A11</font></span>: 
 not  <a href="anproj_9.html#K6" title="ANPROJ_9:func.6">Dir010</a> , <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</a> ,<font color="Maroon" title="c7">P</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/>

<span class="kw">assume </span><a NAME="E2:45_3"/><span class="lab"><font color="Green" title="E36">A12</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:45_3"/><span class="lab"><font color="Green" title="E37">A13</font></span>: 
 not <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="euclid_5.html#T4" onmouseover="rs('euclid_5/T4')" onmouseout="rh()">EUCLID_5:4</a>, <a class="ref" href="finseq_1.html#T78" onmouseover="rs('finseq_1/T78')" onmouseout="rh()">FINSEQ_1:78</a></span>;<br/>
<a NAME="E4:45_3"/><span class="lab"><font color="Green" title="E38">A14</font></span>: 
 not <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>,<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>,<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="anproj_1.html#R2" title="ANPROJ_1:pred.2">are_LinDep</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_3"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="ref" href="anproj_2.html#T23" onmouseover="rs('anproj_2/T23')" onmouseout="rh()">ANPROJ_2:23</a>, <a class="ref" href="anproj_9.html#T10" target="_self" onmouseover="rs('anproj_9/T10')" onmouseout="rh()">Th11</a>, <a class="txt" href="#E3:45_3"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E33">A1</font></span></a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T8" onmouseover="rs('euclid_5/T8')" onmouseout="rh()">EUCLID_5:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">b</font></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T8" onmouseover="rs('euclid_5/T8')" onmouseout="rh()">EUCLID_5:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">b</font></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">c</font></span>)</span></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">a</font></span>)</span>,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">b</font></span>)</span>,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">c</font></span>)</span></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T8" onmouseover="rs('euclid_5/T8')" onmouseout="rh()">EUCLID_5:8</a></span>
<br/>.= 
<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">c</font></span>)</span></span>)</span></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T6" onmouseover="rs('euclid_5/T6')" onmouseout="rh()">EUCLID_5:6</a></span>
<br/>.= 
<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">b</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">c</font></span>)</span></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T6" onmouseover="rs('euclid_5/T6')" onmouseout="rh()">EUCLID_5:6</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 3</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_3"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="ref" href="euclid_5.html#T4" onmouseover="rs('euclid_5/T4')" onmouseout="rh()">EUCLID_5:4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:45_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_3"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="ref" href="anproj_1.html#D2" onmouseover="rs('anproj_1/D2')" onmouseout="rh()">ANPROJ_1:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
