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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"> (  <a href="afproj.html#K13" title="AFPROJ:func.13">IncProjSp_of</a> <font color="Maroon" title="c1">AS</font> is  <a href="incproj.html#V5" title="INCPROJ:attr.5">2-dimensional</a>  implies <font color="Maroon" title="c1">AS</font> is   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</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">assume </span><a NAME="E2:68"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
 <a href="afproj.html#K13" title="AFPROJ:func.13">IncProjSp_of</a> <font color="Maroon" title="c1">AS</font> is  <a href="incproj.html#V5" title="INCPROJ:attr.5">2-dimensional</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   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a></span><br/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:68"/><span class="lab"><font color="Green" title="E52">A2</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">and </span><br/><a NAME="E5:68"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">and </span><br/><a NAME="E6:68"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">and </span><br/><a NAME="E7:68"/><span class="lab"><font color="Green" title="E53">A3</font></span>: 
<font color="Maroon" title="c4">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">assume </span><a NAME="E8:68"/>
<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"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">z</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="E10:68"/><span class="lab"><font color="Green" title="E54">A4</font></span>: 
not <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:68"><span class="lab"><font color="Green" title="E53">A3</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="c6">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="c5">z</font>;<br/>
<a NAME="E11:68"/><span class="lab"><font color="Green" title="E55">A5</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="c5">z</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="#E4:68"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E10:68"><span class="lab"><font color="Green" title="E54">A4</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="c7">A</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K6" title="AFPROJ:func.6">PDir</a> <font color="Maroon" title="c4">X</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c8">K</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<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="c5">z</font></span>)</span>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" title="AFPROJ:func.13">IncProjSp_of</a> <font color="Maroon" title="c1">AS</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:68"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="ref" href="afproj.html#T23" target="_self" title="AFPROJ:th.23">Th23</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">a</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" title="AFPROJ:func.13">IncProjSp_of</a> <font color="Maroon" title="c1">AS</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E14:68"/><span class="lab"><font color="Green" title="E56">A6</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">A</font>
 <span class="kw">and </span><br/><a NAME="E15:68"/><span class="lab"><font color="Green" title="E57">A7</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="incproj.html#D14" title="INCPROJ:def.14">INCPROJ:def 14</a></span>;<br/>
<a NAME="E16:68"/>
<font color="Maroon" title="c9">a</font> is  not   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:68"><span class="lab"><font color="Green" title="E56">A6</font></span></a>, <a class="ref" href="afproj.html#T27" target="_self" title="AFPROJ:th.27">Th27</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">Y9</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="E18:68"/><span class="lab"><font color="Green" title="E58">A8</font></span>: 
<font color="Maroon" title="c9">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="c10">Y9</font>
 <span class="kw">and </span><br/><a NAME="E19:68"/><span class="lab"><font color="Green" title="E59">A9</font></span>: 
<font color="Maroon" title="c10">Y9</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="ref" href="afproj.html#T20" target="_self" title="AFPROJ:th.20">Th20</a></span>;<br/>
<a NAME="E20:68"/>
<font color="Maroon" title="c10">Y9</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</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="c5">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:68"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="txt" href="#E15:68"><span class="lab"><font color="Green" title="E57">A7</font></span></a>, <a class="txt" href="#E18:68"><span class="lab"><font color="Green" title="E58">A8</font></span></a>, <a class="txt" href="#E19:68"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="ref" href="afproj.html#T28" target="_self" title="AFPROJ:th.28">Th28</a></span>;<br/>
<a NAME="E21:68"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A10</font></span>: 
<font color="Maroon" title="c10">Y9</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</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="c5">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:68"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="txt" href="#E19:68"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="ref" href="aff_4.html#T40" title="AFF_4:th.40">AFF_4:40</a></span>;<br/>
<a NAME="E22:68"/><span class="lab"><font color="Green" title="E61">A11</font></span>: 
<font color="Maroon" title="c5">z</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="c5">z</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="E23:68"/><span class="lab"><font color="Green" title="E62">A12</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="c5">z</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="E24:68"/>
<font color="Maroon" title="c10">Y9</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c4">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:68"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E14:68"><span class="lab"><font color="Green" title="E56">A6</font></span></a>, <a class="txt" href="#E18:68"><span class="lab"><font color="Green" title="E58">A8</font></span></a>, <a class="txt" href="#E19:68"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="ref" href="afproj.html#T29" target="_self" title="AFPROJ:th.29">Th29</a></span>;<br/>
<a NAME="E25:68"/><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="c5">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:68"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E7:68"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E11:68"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="txt" href="#E23:68"><span class="lab"><font color="Green" title="E62">A12</font></span></a>, <a class="txt" href="#E21:68"><span class="lab"><font color="Green" title="E60">A10</font></span></a>, <a class="ref" href="aff_4.html#T43" title="AFF_4:th.43">AFF_4:43</a>, <a class="ref" href="aff_4.html#T56" title="AFF_4:th.56">AFF_4:56</a></span>;<br/>
<span class="kw">hence </span><a NAME="E26:68"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:68"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="txt" href="#E22:68"><span class="lab"><font color="Green" title="E61">A11</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/>


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