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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">AS</font> is  not   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a> implies  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#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="b2">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 not <font color="Olive" title="b1">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">A</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:49"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 <span class="kw">and </span><br/><a NAME="E4:49"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 <span class="kw">and </span><br/><a NAME="E5:49"/>
<font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 <span class="kw">and </span><br/><a NAME="E6:49"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
<font color="Maroon" title="c5">X</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_4.html#T37" title="AFF_4:th.37">AFF_4:37</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">A</font> =  <a href="afproj.html#K6" title="AFPROJ:func.6">PDir</a> <font color="Maroon" title="c5">X</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:49"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="afproj.html#T15" target="_self" title="AFPROJ:th.15">Th15</a></span>;<br/>
<span class="kw">assume </span><a NAME="E8:49"/>
<font color="Maroon" title="c1">AS</font> is  not   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not for <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#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><br/> for <font color="Olive" title="b2">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> holds  <font color="Olive" title="b1">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b2">A</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">t</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><span class="kw"> such that </span><br/><a NAME="E10:49"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
not <font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:49"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="aff_4.html#T48" title="AFF_4:th.48">AFF_4:48</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">Y</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c7">t</font>;<br/>
<a NAME="E11:49"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c7">t</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E10:49"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">a</font> =  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c7">t</font></span>)</span> as    <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="afproj.html#T14" target="_self" title="AFPROJ:th.14">Th14</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">a</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not for <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> holds  <font color="Maroon" title="c9">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">A</font></span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c6">A</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c9">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">A</font></span><br/>

<a NAME="E13:49"/><span class="lab"><font color="Green" title="E40">A5</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c7">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/>
<a NAME="E14:49"/><span class="lab"><font color="Green" title="E41">A6</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c7">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></span>;<br/>
<span class="kw">thus </span><a NAME="E15:49"/>
not <font color="Maroon" title="c9">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">A</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:49_1"/>
<font color="Maroon" title="c9">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c6">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:49_1"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c7">t</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c5">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:49"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E11:49"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="ref" href="afproj.html#T36" target="_self" title="AFPROJ:th.36">Th36</a></span>;<br/>
<a NAME="E3:49_1"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c7">t</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E6:49"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E11:49"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E14:49"><span class="lab"><font color="Green" title="E41">A6</font></span></a>, <a class="ref" href="aff_4.html#T43" title="AFF_4:th.43">AFF_4:43</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:49_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:49"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E13:49"><span class="lab"><font color="Green" title="E40">A5</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"> verum</span><br/>


</div><span class="kw">end;</span></div>


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