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<span class="kw">let </span><font color="Maroon" title="c1">CPS</font> be   <a href="anproj_2.html#NM1" title="ANPROJ_2:NM.1">CollProjectiveSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" title="INCPROJ:func.3">IncProjSp_of</a> <font color="Maroon" title="c1">CPS</font></span>)</span>  ex <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" title="INCPROJ:func.3">IncProjSp_of</a> <font color="Maroon" title="c1">CPS</font></span>)</span> st <br/>( <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">c</font> &amp; <font color="Olive" title="b4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b4">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">P</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">P</font> be   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" title="INCPROJ:func.3">IncProjSp_of</a> <font color="Maroon" title="c1">CPS</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" title="INCPROJ:func.3">IncProjSp_of</a> <font color="Maroon" title="c1">CPS</font></span>)</span> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Olive" title="b2">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Olive" title="b3">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">P</font> )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c3">P9</font> = <font color="Maroon" title="c2">P</font> as    <a href="incproj.html#M1" title="INCPROJ:mode.1">LINE</a> of <font color="Maroon" title="c1">CPS</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="incproj.html#T2" target="_self" title="INCPROJ:th.2">Th2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">a99</font>, <font color="Maroon" title="c5">b99</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CPS</font><span class="kw"> such that </span><br/><a NAME="E3:19"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
<font color="Maroon" title="c4">a99</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b99</font>
 <span class="kw">and </span><br/><a NAME="E4:19"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
<font color="Maroon" title="c3">P9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> <font color="Maroon" title="c4">a99</font>,<font color="Maroon" title="c5">b99</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="collsp.html#D7" title="COLLSP:def.7">COLLSP:def 7</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">c9</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CPS</font><span class="kw"> such that </span><br/><a NAME="E6:19"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
( <font color="Maroon" title="c4">a99</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">c9</font> &amp; <font color="Maroon" title="c5">b99</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">c9</font> )
 <span class="kw">and </span><br/><a NAME="E7:19"/><span class="lab"><font color="Green" title="E14">A4</font></span>: 
<font color="Maroon" title="c4">a99</font>,<font color="Maroon" title="c5">b99</font>,<font color="Maroon" title="c6">c9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="anproj_2.html#D10" title="ANPROJ_2:def.10">ANPROJ_2:def 10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c4">a99</font>, <font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c5">b99</font>, <font color="Maroon" title="c9">c</font> = <font color="Maroon" title="c6">c9</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" title="INCPROJ:func.3">IncProjSp_of</a> <font color="Maroon" title="c1">CPS</font></span>)</span> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">a</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">c</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" title="INCPROJ:func.3">IncProjSp_of</a> <font color="Maroon" title="c1">CPS</font></span>)</span> st <br/>( <font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> &amp; <font color="Olive" title="b2">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c7">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Olive" title="b1">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Olive" title="b2">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">P</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c8">b</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">c</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" title="INCPROJ:func.3">IncProjSp_of</a> <font color="Maroon" title="c1">CPS</font></span>)</span> st <br/>( <font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> &amp; <font color="Olive" title="b1">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c7">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c8">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Olive" title="b1">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">P</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c9">c</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c7">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c8">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c9">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">P</font> )</span><br/>

<span class="kw">thus </span><a NAME="E9:19"/>
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E6:19"><span class="lab"><font color="Green" title="E13">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c8">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c9">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">P</font> )</span><br/>

<a NAME="E10:19"/>
<font color="Maroon" title="c4">a99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="collsp.html#T16" title="COLLSP:th.16">COLLSP:16</a></span>;<br/>
<a NAME="E11:19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A5</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incproj.html#T9" target="_self" title="INCPROJ:th.9">Th9</a></span>;<br/>
<a NAME="E12:19"/>
<font color="Maroon" title="c5">b99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="collsp.html#T16" title="COLLSP:th.16">COLLSP:16</a></span>;<br/>
<a NAME="E13:19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A6</font></span>: 
<font color="Maroon" title="c8">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="incproj.html#T9" target="_self" title="INCPROJ:th.9">Th9</a></span>;<br/>
<a NAME="E14:19"/>
 ex <font color="Olive" title="b1">Q</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="incproj.html#K3" title="INCPROJ:func.3">IncProjSp_of</a> <font color="Maroon" title="c1">CPS</font></span>)</span> st <br/>( <font color="Maroon" title="c7">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">Q</font> &amp; <font color="Maroon" title="c8">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">Q</font> &amp; <font color="Maroon" title="c9">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">Q</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:19"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="incproj.html#T14" target="_self" title="INCPROJ:th.14">Th14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:19"/>
( <font color="Maroon" title="c7">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c8">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c9">c</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">P</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E11:19"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="txt" href="#E13:19"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="ref" href="incproj.html#T12" target="_self" title="INCPROJ:th.12">Th12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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