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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"> (  <a href="afproj.html#K14" title="AFPROJ:func.14">ProjHorizon</a> <font color="Maroon" title="c1">AS</font> is   <a href="incproj.html#NM2" title="INCPROJ:NM.2">IncProjSp</a> implies  not <font color="Maroon" title="c1">AS</font> is   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a> )</span><br/>

<b>assume </b><a NAME="E1:70"/><i><font color="Green" title="E53">A1</font></i>: 
 <a href="afproj.html#K14" title="AFPROJ:func.14">ProjHorizon</a> <font color="Maroon" title="c1">AS</font> is   <a href="incproj.html#NM2" title="INCPROJ:NM.2">IncProjSp</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">AS</font> is  not   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a></span><br/>

<b>assume </b><a NAME="E2:70"/><i><font color="Green" title="E54">A2</font></i>: 
<font color="Maroon" title="c1">AS</font> is   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>set </b><font color="Maroon" title="c2">XX</font> =  <a href="afproj.html#K14" title="AFPROJ:func.14">ProjHorizon</a> <font color="Maroon" title="c1">AS</font>;<br/>
<b>consider </b><font color="Maroon" title="c3">a</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>, <font color="Maroon" title="c4">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><b> such that </b><br/><a NAME="E4:70"/><i><font color="Green" title="E55">A3</font></i>: 
not <font color="Maroon" title="c3">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="#E1:70"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="incproj.html#D11" title="INCPROJ:def.11">INCPROJ:def 11</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="E6:70"/><i><font color="Green" title="E56">A4</font></i>: 
( <font color="Maroon" title="c3">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="c5">X</font> &amp; <font color="Maroon" title="c5">X</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_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">Y</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="E8:70"/><i><font color="Green" title="E57">A5</font></i>: 
( <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afproj.html#K6" title="AFPROJ:func.6">PDir</a> <font color="Maroon" title="c6">Y</font> &amp; <font color="Maroon" title="c6">Y</font> is <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> )
 <b>by </b><i><a class="ref" href="afproj.html#T15" target="_self" title="AFPROJ:th.15">Th15</a></i>;<br/>
<a NAME="E9:70"/>
<font color="Maroon" title="c6">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AS</font>
 
<b>by </b><i><a class="txt" href="#E2:70"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="#E8:70"><i><font color="Green" title="E57">A5</font></i></a>, <a class="ref" href="afproj.html#T2" target="_self" title="AFPROJ:th.2">Th2</a></i>;<br/>
<a NAME="E10:70"/><b>then </b>
<font color="Maroon" title="c5">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c6">Y</font>
 
<b>by </b><i><a class="txt" href="#E6:70"><i><font color="Green" title="E56">A4</font></i></a>, <a class="txt" href="#E8:70"><i><font color="Green" title="E57">A5</font></i></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="E11:70"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:70"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="#E6:70"><i><font color="Green" title="E56">A4</font></i></a>, <a class="txt" href="#E8:70"><i><font color="Green" title="E57">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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