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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"> for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>  st <font color="Olive" title="b1">X</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a>  &amp; <font color="Olive" title="b2">Y</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a>  &amp; <font color="Olive" title="b1">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b2">Y</font> holds <br/><font color="Olive" title="b2">Y</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b1">X</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">Y</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>; <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="c2">X</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a>  &amp; <font color="Maroon" title="c3">Y</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a>  &amp; <font color="Maroon" title="c2">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c3">Y</font> implies <font color="Maroon" title="c3">Y</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c2">X</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:74"/><span class="lab"><font color="Green" title="E61">A1</font></span>: 
( <font color="Maroon" title="c2">X</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a>  &amp; <font color="Maroon" title="c3">Y</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a>  )
 <span class="kw">and </span><br/><a NAME="E2:74"/><span class="lab"><font color="Green" title="E62">A2</font></span>: 
<font color="Maroon" title="c2">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c3">Y</font>
 ; <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="c3">Y</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c2">X</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">P</font>, <font color="Maroon" title="c6">M</font>, <font color="Maroon" title="c7">N</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="E4:74"/><span class="lab"><font color="Green" title="E63">A3</font></span>: 
not <font color="Maroon" title="c4">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c5">P</font>
 <span class="kw">and </span><br/><a NAME="E5:74"/><span class="lab"><font color="Green" title="E64">A4</font></span>: 
( <font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c5">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c6">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Maroon" title="c7">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> )
 <span class="kw">and </span><br/><a NAME="E6:74"/><span class="lab"><font color="Green" title="E65">A5</font></span>: 
( <font color="Maroon" title="c4">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c6">M</font> or <font color="Maroon" title="c6">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">A</font> )
 <span class="kw">and </span><br/><a NAME="E7:74"/><span class="lab"><font color="Green" title="E66">A6</font></span>: 
( <font color="Maroon" title="c5">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c7">N</font> or <font color="Maroon" title="c7">N</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c5">P</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="txt" href="#E2:74"><span class="lab"><font color="Green" title="E62">A2</font></span></a>, <a class="ref" href="aff_4.html#T55" target="_self" title="AFF_4:th.55">Th55</a></span>;<br/>
<a NAME="E8:74"/>
not <font color="Maroon" title="c6">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c7">N</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:74_1"/>
<font color="Maroon" title="c6">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c7">N</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="E2:74_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c7">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:74"><span class="lab"><font color="Green" title="E65">A5</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:74_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:74"><span class="lab"><font color="Green" title="E63">A3</font></span></a>, <a class="txt" href="#E7:74"><span class="lab"><font color="Green" title="E66">A6</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</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>
<span class="kw">hence </span><a NAME="E9:74"/>
<font color="Maroon" title="c3">Y</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="txt" href="#E5:74"><span class="lab"><font color="Green" title="E64">A4</font></span></a>, <a class="txt" href="#E6:74"><span class="lab"><font color="Green" title="E65">A5</font></span></a>, <a class="txt" href="#E7:74"><span class="lab"><font color="Green" title="E66">A6</font></span></a>, <a class="ref" href="aff_4.html#T55" target="_self" title="AFF_4:th.55">Th55</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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