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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">L0</font>, <font color="Olive" title="b3">L1</font>, <font color="Olive" title="b4">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><br/> for <font color="Olive" title="b5">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="b2">L0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b5">P</font> &amp; <font color="Olive" title="b3">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b5">P</font> &amp; <font color="Olive" title="b4">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b5">P</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">L0</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">L1</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">L2</font> &amp; <font color="Olive" title="b2">L0</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Olive" title="b4">L2</font> &amp; <font color="Olive" title="b3">L1</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Olive" title="b4">L2</font> holds <br/><font color="Olive" title="b2">L0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">L1</font></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">L0</font>, <font color="Olive" title="b2">L1</font>, <font color="Olive" title="b3">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><br/> for <font color="Olive" title="b4">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">L0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">P</font> &amp; <font color="Olive" title="b2">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">P</font> &amp; <font color="Olive" title="b3">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">P</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">L0</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">L1</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">L2</font> &amp; <font color="Olive" title="b1">L0</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Olive" title="b3">L2</font> &amp; <font color="Olive" title="b2">L1</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Olive" title="b3">L2</font> holds <br/><font color="Olive" title="b1">L0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">L1</font></span><br/><b>let </b><font color="Maroon" title="c3">L0</font>, <font color="Maroon" title="c4">L1</font>, <font color="Maroon" title="c5">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:()"><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="c3">L0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c4">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="c5">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L0</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">L1</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">L2</font> &amp; <font color="Maroon" title="c3">L0</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c5">L2</font> &amp; <font color="Maroon" title="c4">L1</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c5">L2</font> holds <br/><font color="Maroon" title="c3">L0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">L1</font></span><br/><b>let </b><font color="Maroon" title="c6">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="c3">L0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c4">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L0</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">L1</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">L2</font> &amp; <font color="Maroon" title="c3">L0</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c5">L2</font> &amp; <font color="Maroon" title="c4">L1</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c5">L2</font> implies <font color="Maroon" title="c3">L0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">L1</font> )</span><br/>











<b>assume </b><a NAME="E1:121"/><i><font color="Green" title="E94">A1</font></i>: 
( <font color="Maroon" title="c3">L0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c4">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L0</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">L1</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">L2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">L0</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c5">L2</font> or  not <font color="Maroon" title="c4">L1</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c5">L2</font> or <font color="Maroon" title="c3">L0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">L1</font> )</span><br/>

<b>assume </b><a NAME="E2:121"/><i><font color="Green" title="E95">A2</font></i>: 
( <font color="Maroon" title="c3">L0</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c5">L2</font> &amp; <font color="Maroon" title="c4">L1</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c5">L2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">L0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">L1</font></span><br/>

<a NAME="E3:121"/><b>then </b><i><font color="Green" title="E96">A3</font></i>: 
( <font color="Maroon" title="c3">L0</font> <a href="euclid_4.html#NR1" title="EUCLID_4:NR.1">is_line</a>  &amp; <font color="Maroon" title="c4">L1</font> <a href="euclid_4.html#NR1" title="EUCLID_4:NR.1">is_line</a>  &amp; <font color="Maroon" title="c5">L2</font> <a href="euclid_4.html#NR1" title="EUCLID_4:NR.1">is_line</a>  )
 
<b>by </b><i><a class="ref" href="euclidlp.html#T72" target="_self" title="EUCLIDLP:th.72">Th72</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">x0</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="E5:121"/><i><font color="Green" title="E97">A4</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">x0</font> &amp; <font color="Maroon" title="c7">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L0</font> &amp; not <font color="Maroon" title="c7">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">L2</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:121"><i><font color="Green" title="E94">A1</font></i></a>, <a class="txt" href="euclidlp.html#E2:121"><i><font color="Green" title="E95">A2</font></i></a>, <a class="ref" href="euclidlp.html#T86" target="_self" title="EUCLIDLP:th.86">Th86</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">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="E7:121"/><i><font color="Green" title="E98">A5</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">x2</font> &amp; <font color="Maroon" title="c8">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">L2</font> &amp; not <font color="Maroon" title="c8">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">L1</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:121"><i><font color="Green" title="E94">A1</font></i></a>, <a class="txt" href="euclidlp.html#E2:121"><i><font color="Green" title="E95">A2</font></i></a>, <a class="ref" href="euclidlp.html#T86" target="_self" title="EUCLIDLP:th.86">Th86</a></i>;<br/>
<a NAME="E8:121"/><i><font color="Green" title="E99">A6</font></i>: 
<font color="Maroon" title="c3">L0</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="c2">x</font>,<font color="Maroon" title="c7">x0</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E1:121"><i><font color="Green" title="E94">A1</font></i></a>, <a class="txt" href="euclidlp.html#E5:121"><i><font color="Green" title="E97">A4</font></i></a>, <a class="ref" href="euclidlp.html#T69" target="_self" title="EUCLIDLP:th.69">Th69</a></i>;<br/>
<a NAME="E9:121"/><i><font color="Green" title="E100">A7</font></i>: 
<font color="Maroon" title="c7">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font> <a href="euclidlp.html#R4" title="EUCLIDLP:pred.4">_|_</a> <font color="Maroon" title="c8">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E1:121"><i><font color="Green" title="E94">A1</font></i></a>, <a class="txt" href="euclidlp.html#E2:121"><i><font color="Green" title="E95">A2</font></i></a>, <a class="txt" href="euclidlp.html#E5:121"><i><font color="Green" title="E97">A4</font></i></a>, <a class="txt" href="euclidlp.html#E7:121"><i><font color="Green" title="E98">A5</font></i></a>, <a class="ref" href="euclidlp.html#T79" target="_self" title="EUCLIDLP:th.79">Th79</a></i>;<br/>
<a NAME="E10:121"/><b>then </b><i><font color="Green" title="E101">A8</font></i>: 
<font color="Maroon" title="c6">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="c2">x</font>,<font color="Maroon" title="c7">x0</font>,<font color="Maroon" title="c8">x2</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E1:121"><i><font color="Green" title="E94">A1</font></i></a>, <a class="txt" href="euclidlp.html#E5:121"><i><font color="Green" title="E97">A4</font></i></a>, <a class="txt" href="euclidlp.html#E7:121"><i><font color="Green" title="E98">A5</font></i></a>, <a class="ref" href="euclidlp.html#T50" target="_self" title="EUCLIDLP:th.50">Th50</a>, <a class="ref" href="euclidlp.html#T97" target="_self" title="EUCLIDLP:th.97">Th97</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">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><b> such that </b><br/><a NAME="E12:121"/><i><font color="Green" title="E102">A9</font></i>: 
( <font color="Maroon" title="c9">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">L1</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:121"><i><font color="Green" title="E94">A1</font></i></a>, <a class="txt" href="euclidlp.html#E3:121"><i><font color="Green" title="E96">A3</font></i></a>, <a class="ref" href="euclidlp.html#T58" target="_self" title="EUCLIDLP:th.58">Th58</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">a1</font>, <font color="Maroon" title="c11">a2</font>, <font color="Maroon" title="c12">a3</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E14:121"/><i><font color="Green" title="E103">A10</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c11">a2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c12">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">a1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">x0</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">a3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">x2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:121"><i><font color="Green" title="E94">A1</font></i></a>, <a class="txt" href="euclidlp.html#E10:121"><i><font color="Green" title="E101">A8</font></i></a>, <a class="txt" href="euclidlp.html#E12:121"><i><font color="Green" title="E102">A9</font></i></a>, <a class="ref" href="euclidlp.html#T93" target="_self" title="EUCLIDLP:th.93">Th93</a></i>;<br/>
<i><font color="Green" title="E104">A11</font></i>: <font color="Maroon" title="c9">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font> = 
<span class="p1">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">a2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">a3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E14:121"><i><font color="Green" title="E103">A10</font></i></a>, <a class="ref" href="euclidlp.html#T32" target="_self" title="EUCLIDLP:th.32">Th32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">a3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T32" title="RVSUM_1:th.32">RVSUM_1:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">a2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">a3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T32" title="RVSUM_1:th.32">RVSUM_1:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">a2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">a3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T7" target="_self" title="EUCLIDLP:th.7">Th7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">a3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T1" title="EUCLID_4:th.1">EUCLID_4:1</a></i>
;<br/>
<a NAME="E16:121"/><i><font color="Green" title="E105">A12</font></i>: 
<font color="Maroon" title="c4">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="c2">x</font>,<font color="Maroon" title="c9">x1</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E1:121"><i><font color="Green" title="E94">A1</font></i></a>, <a class="txt" href="euclidlp.html#E12:121"><i><font color="Green" title="E102">A9</font></i></a>, <a class="ref" href="euclidlp.html#T69" target="_self" title="EUCLIDLP:th.69">Th69</a></i>;<br/>
<a NAME="E17:121"/><i><font color="Green" title="E106">A13</font></i>: 
<font color="Maroon" title="c9">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font> <a href="euclidlp.html#R4" title="EUCLIDLP:pred.4">_|_</a> <font color="Maroon" title="c8">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E1:121"><i><font color="Green" title="E94">A1</font></i></a>, <a class="txt" href="euclidlp.html#E2:121"><i><font color="Green" title="E95">A2</font></i></a>, <a class="txt" href="euclidlp.html#E7:121"><i><font color="Green" title="E98">A5</font></i></a>, <a class="txt" href="euclidlp.html#E12:121"><i><font color="Green" title="E102">A9</font></i></a>, <a class="ref" href="euclidlp.html#T79" target="_self" title="EUCLIDLP:th.79">Th79</a></i>;<br/>
<a NAME="E18:121"/>
<font color="Maroon" title="c7">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c8">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font> <a href="euclid_4.html#R1" title="EUCLID_4:pred.1">are_orthogonal</a> 
 
<b>by </b><i><a class="txt" href="euclidlp.html#E9:121"><i><font color="Green" title="E100">A7</font></i></a>, <a class="ref" href="euclidlp.html#D3" target="_self" title="EUCLIDLP:def.3">Def3</a></i>;<br/>
<a NAME="E19:121"/><b>then </b><i><font color="Green" title="E107">A14</font></i>: 
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="euclid_4.html#D6" title="EUCLID_4:def.6">EUCLID_4:def 6</a></i>;<br/>
<a NAME="E20:121"/><i><font color="Green" title="E108">A15</font></i>: 
<font color="Maroon" title="c9">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c8">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font> <a href="euclid_4.html#R1" title="EUCLID_4:pred.1">are_orthogonal</a> 
 
<b>by </b><i><a class="txt" href="euclidlp.html#E17:121"><i><font color="Green" title="E106">A13</font></i></a>, <a class="ref" href="euclidlp.html#D3" target="_self" title="EUCLIDLP:def.3">Def3</a></i>;<br/>
<a NAME="E21:121"/>
<font color="Maroon" title="c8">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E7:121"><i><font color="Green" title="E98">A5</font></i></a>, <a class="ref" href="euclidlp.html#T14" target="_self" title="EUCLIDLP:th.14">Th14</a></i>;<br/>
<a NAME="E22:121"/><b>then </b><i><font color="Green" title="E109">A16</font></i>: 
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="euclid_4.html#T22" title="EUCLID_4:th.22">EUCLID_4:22</a></i>;<br/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">a3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
<b>by </b><i><a class="txt" href="euclidlp.html#E15:121"><i><font color="Green" title="E104">A11</font></i></a>, <a class="txt" href="euclidlp.html#E20:121"><i><font color="Green" title="E108">A15</font></i></a>, <a class="ref" href="euclid_4.html#D6" title="EUCLID_4:def.6">EUCLID_4:def 6</a></i>
<br/>.= 
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">a2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">a3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_4.html#T25" title="EUCLID_4:th.25">EUCLID_4:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c8">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">a3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_4.html#T26" title="EUCLID_4:th.26">EUCLID_4:26</a></i>
<br/>.= 
<font color="Maroon" title="c12">a3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
<b>by </b><i><a class="txt" href="euclidlp.html#E19:121"><i><font color="Green" title="E107">A14</font></i></a>, <a class="ref" href="euclid_4.html#T26" title="EUCLID_4:th.26">EUCLID_4:26</a></i>
;<br/>
<b>then </b><i><font color="Green" title="E110">A17</font></i>: <font color="Maroon" title="c9">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E15:121"><i><font color="Green" title="E104">A11</font></i></a>, <a class="txt" href="euclidlp.html#E22:121"><i><font color="Green" title="E109">A16</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T3" title="EUCLID_4:th.3">EUCLID_4:3</a></i>
<br/>.= 
<font color="Maroon" title="c11">a2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T1" title="EUCLID_4:th.1">EUCLID_4:1</a></i>
;<br/>
<a NAME="E25:121"/>
( <font color="Maroon" title="c9">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c7">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E5:121"><i><font color="Green" title="E97">A4</font></i></a>, <a class="txt" href="euclidlp.html#E12:121"><i><font color="Green" title="E102">A9</font></i></a>, <a class="ref" href="euclidlp.html#T14" target="_self" title="EUCLIDLP:th.14">Th14</a></i>;<br/>
<a NAME="E26:121"/><b>then </b>
<font color="Maroon" title="c9">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font> <a href="euclidlp.html#R2" title="EUCLIDLP:pred.2">//</a> <font color="Maroon" title="c7">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E24:121"><i><font color="Green" title="E110">A17</font></i></a>, <a class="ref" href="euclidlp.html#D1" target="_self" title="EUCLIDLP:def.1">Def1</a></i>;<br/>
<a NAME="E27:121"/><b>then </b><i><font color="Green" title="E111">A18</font></i>: 
<font color="Maroon" title="c4">L1</font> <a href="euclidlp.html#R5" title="EUCLIDLP:pred.5">//</a> <font color="Maroon" title="c3">L0</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E8:121"><i><font color="Green" title="E99">A6</font></i></a>, <a class="txt" href="euclidlp.html#E16:121"><i><font color="Green" title="E105">A12</font></i></a>, <a class="ref" href="euclidlp.html#D7" target="_self" title="EUCLIDLP:def.7">Def7</a></i>;<br/>
<a NAME="E28:121"/>
<font color="Maroon" title="c4">L1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">L0</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E1:121"><i><font color="Green" title="E94">A1</font></i></a>, <a class="ref" href="euclidlp.html#T54" target="_self" title="EUCLIDLP:th.54">Th54</a></i>;<br/>
<b>hence </b><a NAME="E29:121"/>
<font color="Maroon" title="c3">L0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">L1</font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E27:121"><i><font color="Green" title="E111">A18</font></i></a>, <a class="ref" href="euclidlp.html#T76" target="_self" title="EUCLIDLP:th.76">Th76</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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