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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">r</font>, <font color="Olive" title="b4">r1</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CPS</font> st <br/>for <font color="Olive" title="b5">s</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CPS</font>  holds <br/>(  not <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">p1</font>,<font color="Olive" title="b5">s</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  or  not <font color="Olive" title="b3">r</font>,<font color="Olive" title="b4">r1</font>,<font color="Olive" title="b5">s</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  ) implies  ex <font color="Olive" title="b1">M</font>, <font color="Olive" title="b2">N</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/>for <font color="Olive" title="b3">q</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>  holds <br/>(  not <font color="Olive" title="b3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">M</font> or  not <font color="Olive" title="b3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">N</font> ) )</span><br/>

<b>given </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">r</font>, <font color="Maroon" title="c5">r1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CPS</font><b> such that </b><a NAME="E2:18"/><i><font color="Green" title="E14">A1</font></i>: 
for <font color="Olive" title="b1">s</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CPS</font>  holds <br/>(  not <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">p1</font>,<font color="Olive" title="b1">s</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  or  not <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">r1</font>,<font color="Olive" title="b1">s</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">  ex <font color="Olive" title="b1">M</font>, <font color="Olive" title="b2">N</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/>for <font color="Olive" title="b3">q</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>  holds <br/>(  not <font color="Olive" title="b3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">M</font> or  not <font color="Olive" title="b3">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">N</font> )</span><br/>

<b>set </b><font color="Maroon" title="c6">M''</font> =  <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">p1</font>;<br/>
<b>set </b><font color="Maroon" title="c7">N''</font> =  <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">r1</font>;<br/>
<a NAME="E3:18"/>
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p1</font> &amp; <font color="Maroon" title="c4">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">r1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1"/><i><font color="Green" title="E15">A2</font></i>: 
(  not <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p1</font> or  not <font color="Maroon" title="c4">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">r1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<div><i><font color="Green" title="E16">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_1"/>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:18_1_1"/><b>then </b>
( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">r1</font>,<font color="Maroon" title="c5">r1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="collsp.html#T7" title="COLLSP:th.7">COLLSP:7</a></i>;<br/><b>hence </b><a NAME="E3:18_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:18"><i><font color="Green" title="E14">A1</font></i></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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_2"/>
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:18_1_2"/><b>then </b>
( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c3">p1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">r1</font>,<font color="Maroon" title="c3">p1</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <b>by </b><i><a class="ref" href="collsp.html#T7" title="COLLSP:th.7">COLLSP:7</a></i>;<br/><b>hence </b><a NAME="E3:18_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:18"><i><font color="Green" title="E14">A1</font></i></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>hence </b><a NAME="E4:18_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:18_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="#E2:18_1"><i><font color="Green" title="E16">A3</font></i></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>then reconsider </b><font color="Maroon" title="c8">M'</font> =  <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c9">N'</font> =  <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">r1</font> as    <a href="incproj.html#M1" title="INCPROJ:mode.1">LINE</a> of <font color="Maroon" title="c1">CPS</font> <b>by </b><i><a class="ref" href="collsp.html#D7" title="COLLSP:def.7">COLLSP:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">M</font> = <font color="Maroon" title="c8">M'</font>, <font color="Maroon" title="c11">N</font> = <font color="Maroon" title="c9">N'</font> as   <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> <b>by </b><i><a class="ref" href="incproj.html#T2" target="_self" title="INCPROJ:th.2">Th2</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">M</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">N</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/>for <font color="Olive" title="b2">q</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>  holds <br/>(  not <font color="Olive" title="b2">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">M</font> or  not <font color="Olive" title="b2">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">N</font> )</span><br/>

<b>take </b>
<font color="Maroon" title="c11">N</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">q</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>  holds <br/>(  not <font color="Olive" title="b1">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">M</font> or  not <font color="Olive" title="b1">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">N</font> )</span><br/>

<b>thus </b><a NAME="E6:18"/>
for <font color="Olive" title="b1">q</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>  holds <br/>(  not <font color="Olive" title="b1">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">M</font> or  not <font color="Olive" title="b1">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">N</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2"/>
 ex <font color="Olive" title="b1">q</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">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Olive" title="b1">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">N</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c12">q</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><b> such that </b><br/><a NAME="E3:18_2"/><i><font color="Green" title="E15">A4</font></i>: 
( <font color="Maroon" title="c12">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c10">M</font> &amp; <font color="Maroon" title="c12">q</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c11">N</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c13">q'</font> = <font color="Maroon" title="c12">q</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CPS</font> ;<br/>
<a NAME="E5:18_2"/>
( <font color="Maroon" title="c13">q'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">p1</font> &amp; <font color="Maroon" title="c13">q'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="collsp.html#K1" title="COLLSP:func.1">Line</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">r1</font> )
 
<b>by </b><i><a class="txt" href="#E3:18_2"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="incproj.html#T9" target="_self" title="INCPROJ:th.9">Th9</a></i>;<br/>
<a NAME="E6:18_2"/><b>then </b>
( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c13">q'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">r1</font>,<font color="Maroon" title="c13">q'</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 
<b>by </b><i><a class="ref" href="collsp.html#T17" title="COLLSP:th.17">COLLSP:17</a></i>;<br/>
<b>hence </b><a NAME="E7:18_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:18"><i><font color="Green" title="E14">A1</font></i></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>


</div>
