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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">x</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b2">L</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="euclidlp.html#K1" title="EUCLIDLP:func.1">line_of_REAL</a> <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b3">P</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="euclidlp.html#K5" title="EUCLIDLP:func.5">plane_of_REAL</a> <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">P</font> &amp; <font color="Olive" title="b2">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">P</font> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">L</font> &amp; <font color="Olive" title="b2">L</font> is <a href="euclid_4.html#V1" title="EUCLID_4:attr.1">being_line</a> holds <br/><font color="Olive" title="b3">P</font> is <a href="euclidlp.html#V1" title="EUCLIDLP:attr.1">being_plane</a></span><br/><b>let </b><font color="Maroon" title="c2">x</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>; <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">L</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="euclidlp.html#K1" title="EUCLIDLP:func.1">line_of_REAL</a> <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b2">P</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="euclidlp.html#K5" title="EUCLIDLP:func.5">plane_of_REAL</a> <font color="Maroon" title="c1">n</font>  st <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b1">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">P</font> &amp; not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">L</font> &amp; <font color="Olive" title="b1">L</font> is <a href="euclid_4.html#V1" title="EUCLID_4:attr.1">being_line</a> holds <br/><font color="Olive" title="b2">P</font> is <a href="euclidlp.html#V1" title="EUCLIDLP:attr.1">being_plane</a></span><br/><b>let </b><font color="Maroon" title="c3">L</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="euclidlp.html#K1" title="EUCLIDLP:func.1">line_of_REAL</a> <font color="Maroon" title="c1">n</font>; <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">P</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="euclidlp.html#K5" title="EUCLIDLP:func.5">plane_of_REAL</a> <font color="Maroon" title="c1">n</font>  st <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c3">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> &amp; not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c3">L</font> is <a href="euclid_4.html#V1" title="EUCLID_4:attr.1">being_line</a> holds <br/><font color="Olive" title="b1">P</font> is <a href="euclidlp.html#V1" title="EUCLIDLP:attr.1">being_plane</a></span><br/><b>let </b><font color="Maroon" title="c4">P</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="euclidlp.html#K5" title="EUCLIDLP:func.5">plane_of_REAL</a> <font color="Maroon" title="c1">n</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c3">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">P</font> &amp; not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c3">L</font> is <a href="euclid_4.html#V1" title="EUCLID_4:attr.1">being_line</a> implies <font color="Maroon" title="c4">P</font> is <a href="euclidlp.html#V1" title="EUCLIDLP:attr.1">being_plane</a> )</span><br/>







<b>assume </b><a NAME="E1:114"/><i><font color="Green" title="E90">A1</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c3">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">P</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L</font> or  not <font color="Maroon" title="c3">L</font> is <a href="euclid_4.html#V1" title="EUCLID_4:attr.1">being_line</a> or <font color="Maroon" title="c4">P</font> is <a href="euclidlp.html#V1" title="EUCLIDLP:attr.1">being_plane</a> )</span><br/>

<b>assume </b><a NAME="E2:114"/>
( not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c3">L</font> is <a href="euclid_4.html#V1" title="EUCLID_4:attr.1">being_line</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">P</font> is <a href="euclidlp.html#V1" title="EUCLIDLP:attr.1">being_plane</a></span><br/>

<b>then consider </b><font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c6">x2</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E4:114"/><i><font color="Green" title="E91">A2</font></i>: 
( <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font> &amp; <font color="Maroon" title="c2">x</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c5">x1</font> <a href="euclidlp.html#R3" title="EUCLIDLP:pred.3">are_lindependent2</a>  )
 <b>by </b><i><a class="ref" href="euclidlp.html#T60" target="_self" title="EUCLIDLP:th.60">Th60</a></i>;<br/>
<a NAME="E5:114"/>
( <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L</font> )
 
<b>by </b><i><a class="txt" href="#E4:114"><i><font color="Green" title="E91">A2</font></i></a>, <a class="ref" href="euclid_4.html#T10" title="EUCLID_4:th.10">EUCLID_4:10</a></i>;<br/>
<a NAME="E6:114"/><b>then </b>
<font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">x2</font>
 
<b>by </b><i><a class="txt" href="#E1:114"><i><font color="Green" title="E90">A1</font></i></a>, <a class="txt" href="#E4:114"><i><font color="Green" title="E91">A2</font></i></a>, <a class="ref" href="euclidlp.html#T97" target="_self" title="EUCLIDLP:th.97">Th97</a></i>;<br/>
<b>hence </b><a NAME="E7:114"/>
<font color="Maroon" title="c4">P</font> is <a href="euclidlp.html#V1" title="EUCLIDLP:attr.1">being_plane</a>
 <b>by </b><i><a class="txt" href="#E4:114"><i><font color="Green" title="E91">A2</font></i></a>, <a class="ref" href="euclidlp.html#D10" target="_self" title="EUCLIDLP:def.10">Def10</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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