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<b>let </b><font color="Maroon" title="c1">AS</font> be   <a href="diraf.html#NM1" title="DIRAF:NM.1">AffinSpace</a>; <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">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="afproj.html#K14" title="AFPROJ:func.14">ProjHorizon</a> <font color="Maroon" title="c1">AS</font></span>)</span>  ex <font color="Olive" title="b3">A</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="afproj.html#K14" title="AFPROJ:func.14">ProjHorizon</a> <font color="Maroon" title="c1">AS</font></span>)</span> st <br/>( <font color="Olive" title="b1">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">A</font> &amp; <font color="Olive" title="b2">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">A</font> )</span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <span class="p1">(<span class="default"><a href="afproj.html#K14" title="AFPROJ:func.14">ProjHorizon</a> <font color="Maroon" title="c1">AS</font></span>)</span>; <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">A</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="afproj.html#K14" title="AFPROJ:func.14">ProjHorizon</a> <font color="Maroon" title="c1">AS</font></span>)</span> st <br/>( <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">A</font> &amp; <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">A</font> )</span><br/>



<b>consider </b><font color="Maroon" title="c4">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E2:48"/><i><font color="Green" title="E37">A1</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c4">X</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 <b>by </b><i><a class="ref" href="afproj.html#T14" target="_self" title="AFPROJ:th.14">Th14</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">X'</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E4:48"/><i><font color="Green" title="E38">A2</font></i>: 
( <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Maroon" title="c5">X'</font> &amp; <font color="Maroon" title="c5">X'</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 <b>by </b><i><a class="ref" href="afproj.html#T14" target="_self" title="AFPROJ:th.14">Th14</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E6:48"/><i><font color="Green" title="E39">A3</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X'</font>
 <b>and </b><br/><a NAME="E7:48"/>
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X'</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">y</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E4:48"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="aff_1.html#T31" title="AFF_1:th.31">AFF_1:31</a></i>;<br/>
<a NAME="E8:48"/><i><font color="Green" title="E40">A4</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c4">X</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c4">X</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c6">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c4">X</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E2:48"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="aff_4.html#T27" title="AFF_4:th.27">AFF_4:27</a>, <a class="ref" href="aff_4.html#D3" title="AFF_4:def.3">AFF_4:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">Z</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E10:48"/><i><font color="Green" title="E41">A5</font></i>: 
( <font color="Maroon" title="c5">X'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Z</font> &amp; <font color="Maroon" title="c6">x</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Z</font> &amp; <font color="Maroon" title="c8">Z</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  )
 <b>by </b><i><a class="txt" href="afproj.html#E4:48"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="afproj.html#E6:48"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="aff_4.html#T38" title="AFF_4:th.38">AFF_4:38</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">A</font> =  <a href="afproj.html#K6" title="AFPROJ:func.6">PDir</a> <font color="Maroon" title="c8">Z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <span class="p1">(<span class="default"><a href="afproj.html#K14" title="AFPROJ:func.14">ProjHorizon</a> <font color="Maroon" title="c1">AS</font></span>)</span> <b>by </b><i><a class="txt" href="afproj.html#E10:48"><i><font color="Green" title="E41">A5</font></i></a>, <a class="ref" href="afproj.html#T15" target="_self" title="AFPROJ:th.15">Th15</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">A</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">A</font> )</span><br/>

<a NAME="E12:48"/>
( <font color="Maroon" title="c4">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c8">Z</font> &amp; <font color="Maroon" title="c5">X'</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c8">Z</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E2:48"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="afproj.html#E4:48"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="afproj.html#E8:48"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="afproj.html#E10:48"><i><font color="Green" title="E41">A5</font></i></a>, <a class="ref" href="aff_4.html#T41" title="AFF_4:th.41">AFF_4:41</a>, <a class="ref" href="aff_4.html#T42" title="AFF_4:th.42">AFF_4:42</a></i>;<br/>
<b>hence </b><a NAME="E13:48"/>
( <font color="Maroon" title="c2">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c3">b</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c9">A</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E2:48"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="afproj.html#E4:48"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="afproj.html#E10:48"><i><font color="Green" title="E41">A5</font></i></a>, <a class="ref" href="afproj.html#T36" target="_self" title="AFPROJ:th.36">Th36</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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