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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">L1</font>, <font color="Olive" title="b2">L2</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>  st <font color="Olive" title="b1">L1</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Olive" title="b2">L2</font> &amp; <font color="Olive" title="b1">L1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b2">L2</font> holds <br/> ex <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 <br/>( <font color="Olive" title="b3">P</font> is  <a href="euclidlp.html#V1" title="EUCLIDLP:attr.1">being_plane</a>  &amp; <font color="Olive" title="b1">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">P</font> &amp; <font color="Olive" title="b2">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">P</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">L1</font>, <font color="Maroon" title="c3">L2</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">L1</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c3">L2</font> &amp; <font color="Maroon" title="c2">L1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">L2</font> implies  ex <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 <br/>( <font color="Olive" title="b1">P</font> is  <a href="euclidlp.html#V1" title="EUCLIDLP:attr.1">being_plane</a>  &amp; <font color="Maroon" title="c2">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c3">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:120"/><span class="lab"><font color="Green" title="E92">A1</font></span>: 
<font color="Maroon" title="c2">L1</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c3">L2</font>
 <span class="kw">and </span><br/><a NAME="E2:120"/><span class="lab"><font color="Green" title="E93">A2</font></span>: 
<font color="Maroon" title="c2">L1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">L2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <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 <br/>( <font color="Olive" title="b1">P</font> is  <a href="euclidlp.html#V1" title="EUCLIDLP:attr.1">being_plane</a>  &amp; <font color="Maroon" title="c2">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c3">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">x1</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><span class="kw"> such that </span><br/><a NAME="E4:120"/><span class="lab"><font color="Green" title="E94">A3</font></span>: 
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L1</font>
 <span class="kw">and </span><br/><a NAME="E5:120"/><span class="lab"><font color="Green" title="E95">A4</font></span>: 
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:120"><span class="lab"><font color="Green" title="E93">A2</font></span></a>, <a class="ref" href="euclidlp.html#T49" target="_self" title="EUCLIDLP:th.49">Th54</a></span>;<br/>
<a NAME="E6:120"/>
<font color="Maroon" title="c2">L1</font> is  <a href="euclid_4.html#V1" title="EUCLID_4:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:120"><span class="lab"><font color="Green" title="E92">A1</font></span></a>, <a class="ref" href="euclidlp.html#T67" target="_self" title="EUCLIDLP:th.67">Th72</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">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><span class="kw"> such that </span><br/><a NAME="E8:120"/><span class="lab"><font color="Green" title="E96">A5</font></span>: 
( <font color="Maroon" title="c5">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x1</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T53" target="_self" title="EUCLIDLP:th.53">Th58</a></span>;<br/>
<a NAME="E9:120"/><span class="lab"><font color="Green" title="E97">A6</font></span>: 
<font color="Maroon" title="c2">L1</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="c4">x1</font>,<font color="Maroon" title="c5">x2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:120"><span class="lab"><font color="Green" title="E94">A3</font></span></a>, <a class="txt" href="#E8:120"><span class="lab"><font color="Green" title="E96">A5</font></span></a>, <a class="ref" href="euclidlp.html#T64" target="_self" title="EUCLIDLP:th.64">Th69</a></span>;<br/>
<a NAME="E10:120"/>
<font color="Maroon" title="c3">L2</font> is  <a href="euclid_4.html#V1" title="EUCLID_4:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:120"><span class="lab"><font color="Green" title="E92">A1</font></span></a>, <a class="ref" href="euclidlp.html#T67" target="_self" title="EUCLIDLP:th.67">Th72</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x3</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><span class="kw"> such that </span><br/><a NAME="E12:120"/><span class="lab"><font color="Green" title="E98">A7</font></span>: 
( <font color="Maroon" title="c6">x3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x1</font> &amp; <font color="Maroon" title="c6">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T53" target="_self" title="EUCLIDLP:th.53">Th58</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">P</font> =  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> (<font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">x3</font>) as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">P</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">P</font> is    <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> &amp; <font color="Maroon" title="c7">P</font> is  <a href="euclidlp.html#V1" title="EUCLIDLP:attr.1">being_plane</a>  &amp; <font color="Maroon" title="c2">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">P</font> &amp; <font color="Maroon" title="c3">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">P</font> )</span><br/>

<a NAME="E14:120"/><span class="lab"><font color="Green" title="E99">A8</font></span>: 
( <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T82" target="_self" title="EUCLIDLP:th.82">Th87</a></span>;<br/>
<a NAME="E15:120"/><span class="lab"><font color="Green" title="E100">A9</font></span>: 
<font color="Maroon" title="c6">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T82" target="_self" title="EUCLIDLP:th.82">Th87</a></span>;<br/>
<a NAME="E16:120"/><span class="lab"><font color="Green" title="E101">A10</font></span>: 
<font color="Maroon" title="c3">L2</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="c4">x1</font>,<font color="Maroon" title="c6">x3</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:120"><span class="lab"><font color="Green" title="E95">A4</font></span></a>, <a class="txt" href="#E12:120"><span class="lab"><font color="Green" title="E98">A7</font></span></a>, <a class="ref" href="euclidlp.html#T64" target="_self" title="EUCLIDLP:th.64">Th69</a></span>;<br/>
<a NAME="E17:120"/>
<font color="Maroon" title="c5">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">x1</font> <a href="euclidlp.html#R3" title="EUCLIDLP:pred.3">are_lindependent2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:120"><span class="lab"><font color="Green" title="E92">A1</font></span></a>, <a class="txt" href="#E4:120"><span class="lab"><font color="Green" title="E94">A3</font></span></a>, <a class="txt" href="#E5:120"><span class="lab"><font color="Green" title="E95">A4</font></span></a>, <a class="txt" href="#E8:120"><span class="lab"><font color="Green" title="E96">A5</font></span></a>, <a class="txt" href="#E12:120"><span class="lab"><font color="Green" title="E98">A7</font></span></a>, <a class="ref" href="euclidlp.html#T45" target="_self" title="EUCLIDLP:th.45">Th50</a>, <a class="ref" href="euclidlp.html#T74" target="_self" title="EUCLIDLP:th.74">Th79</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:120"/>
( <font color="Maroon" title="c7">P</font> is    <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> &amp; <font color="Maroon" title="c7">P</font> is  <a href="euclidlp.html#V1" title="EUCLIDLP:attr.1">being_plane</a>  &amp; <font color="Maroon" title="c2">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">P</font> &amp; <font color="Maroon" title="c3">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">P</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:120"><span class="lab"><font color="Green" title="E97">A6</font></span></a>, <a class="txt" href="#E16:120"><span class="lab"><font color="Green" title="E101">A10</font></span></a>, <a class="txt" href="#E14:120"><span class="lab"><font color="Green" title="E99">A8</font></span></a>, <a class="txt" href="#E15:120"><span class="lab"><font color="Green" title="E100">A9</font></span></a>, <a class="ref" href="euclidlp.html#D10" target="_self" title="EUCLIDLP:def.10">Def10</a>, <a class="ref" href="euclidlp.html#T85" target="_self" title="EUCLIDLP:th.85">Th90</a>, <a class="ref" href="euclidlp.html#T90" target="_self" title="EUCLIDLP:th.90">Th95</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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