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<span class="kw">let </span><font color="Maroon" title="c4">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"> (  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="c4">P</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</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="c4">P</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</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="c4">P</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a>  implies  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   non  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  <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> st <font color="Maroon" title="c4">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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:46"/><span class="lab"><font color="Green" title="E34">A1</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="c4">P</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E2:46"/><span class="lab"><font color="Green" title="E35">A2</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="c4">P</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E3:46"/><span class="lab"><font color="Green" title="E36">A3</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="c4">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">  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   non  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  <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> st <font color="Maroon" title="c4">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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font> being   <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><span class="kw"> such that </span><br/><a NAME="E5:46"/><span class="lab"><font color="Green" title="E37">A4</font></span>: 
<font color="Maroon" title="c4">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="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>
 <span class="kw">and </span><br/><a NAME="E6:46"/><span class="lab"><font color="Green" title="E38">A5</font></span>: 
( <font color="Maroon" title="c5">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="c6">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="c7">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 class="kw">by</span> <span class="lab"><a class="ref" href="anproj_9.html#T26" target="_self" onmouseover="rs('anproj_9/T26')" onmouseout="rh()">Th22</a></span>;<br/>
<a NAME="E7:46"/>
(  not <font color="Maroon" title="c5">a</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  not <font color="Maroon" title="c6">b</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  not <font color="Maroon" title="c7">c</font> 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="#E1:46"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="txt" href="#E6:46"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="txt" href="anproj_9.html#E45"><span class="lab"><font color="Green" title="E32">Lem07</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">a</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c9">b</font> = <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c10">c</font> = <font color="Maroon" title="c7">c</font> as   non  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  <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> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">a</font>
; <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">  ex <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">c</font> being   non  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  <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> st <font color="Maroon" title="c4">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="c8">a</font>,<font color="Olive" title="b1">b</font>,<font color="Olive" title="b2">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c9">b</font>
; <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">  ex <font color="Olive" title="b1">c</font> being   non  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  <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> st <font color="Maroon" title="c4">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="c8">a</font>,<font color="Maroon" title="c9">b</font>,<font color="Olive" title="b1">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c10">c</font>
; <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">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="c8">a</font>,<font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c10">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span><br/>

<span class="kw">thus </span><a NAME="E9:46"/>
<font color="Maroon" title="c4">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="c8">a</font>,<font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c10">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="txt" href="#E5:46"><span class="lab"><font color="Green" title="E37">A4</font></span></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/>


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