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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> holds <br/><font color="Olive" title="b1">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">L2</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> implies <font color="Maroon" title="c2">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">L2</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:83"/><span class="lab"><font color="Green" title="E68">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>
 ; <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="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">L2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:83_1_1"/>
( <font color="Maroon" title="c2">L1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">L2</font> or <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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1"><span class="kw">case </span></a><a NAME="E1:83_1_1_1"/><span class="lab"><font color="Green" title="E69">A2</font></span>: 
<font color="Maroon" title="c2">L1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</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"> <font color="Maroon" title="c2">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">L2</font></span><br/><div class="add"><a NAME="E2:83_1_1_1"/>
 ex <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> st <font color="Olive" title="b1">x</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#T52" target="_self" title="EUCLIDLP:th.52">Th57</a></span>;<br/><span class="kw">hence </span><a NAME="E3:83_1_1_1"/>
<font color="Maroon" title="c2">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">L2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_1_1_1"><span class="lab"><font color="Green" title="E69">A2</font></span></a>, <a class="ref" href="euclidlp.html#T49" target="_self" title="EUCLIDLP:th.49">Th54</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_2"><span class="kw">case </span></a><a NAME="E1:83_1_1_2"/>
<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"> <font color="Maroon" title="c2">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">L2</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">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><span class="kw"> such that </span><br/><a NAME="E3:83_1_1_2"/><span class="lab"><font color="Green" title="E69">A3</font></span>: 
<font color="Maroon" title="c4">x0</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="E4:83_1_1_2"/><span class="lab"><font color="Green" title="E70">A4</font></span>: 
<font color="Maroon" title="c4">x0</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#T49" target="_self" title="EUCLIDLP:th.49">Th54</a></span>;<br/><a NAME="E5:83_1_1_2"/>
<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="#E1:83"><span class="lab"><font color="Green" title="E68">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">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="E7:83_1_1_2"/><span class="lab"><font color="Green" title="E71">A5</font></span>: 
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font>
 <span class="kw">and </span><br/><a NAME="E8:83_1_1_2"/><span class="lab"><font color="Green" title="E72">A6</font></span>: 
<font color="Maroon" title="c5">x1</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="E9:83_1_1_2"/><span class="lab"><font color="Green" title="E73">A7</font></span>: 
<font color="Maroon" title="c5">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">x0</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83_1_1_2"><span class="lab"><font color="Green" title="E71">A5</font></span></a>, <a class="ref" href="euclidlp.html#T9" target="_self" title="EUCLIDLP:th.9">Th14</a></span>;<br/><a NAME="E10:83_1_1_2"/>
<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="#E1:83"><span class="lab"><font color="Green" title="E68">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">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="E12:83_1_1_2"/><span class="lab"><font color="Green" title="E74">A8</font></span>: 
<font color="Maroon" title="c6">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font>
 <span class="kw">and </span><br/><a NAME="E13:83_1_1_2"/><span class="lab"><font color="Green" title="E75">A9</font></span>: 
<font color="Maroon" title="c6">x2</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/><a NAME="E14:83_1_1_2"/><span class="lab"><font color="Green" title="E76">A10</font></span>: 
<font color="Maroon" title="c6">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">x0</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:83_1_1_2"><span class="lab"><font color="Green" title="E74">A8</font></span></a>, <a class="ref" href="euclidlp.html#T9" target="_self" title="EUCLIDLP:th.9">Th14</a></span>;<br/><a NAME="E15:83_1_1_2"/><span class="lab"><font color="Green" title="E77">A11</font></span>: 
<font color="Maroon" title="c5">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">x0</font> <a href="euclidlp.html#R4" title="EUCLIDLP:pred.4">_|_</a> <font color="Maroon" title="c6">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="txt" href="#E3:83_1_1_2"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="txt" href="#E4:83_1_1_2"><span class="lab"><font color="Green" title="E70">A4</font></span></a>, <a class="txt" href="#E7:83_1_1_2"><span class="lab"><font color="Green" title="E71">A5</font></span></a>, <a class="txt" href="#E8:83_1_1_2"><span class="lab"><font color="Green" title="E72">A6</font></span></a>, <a class="txt" href="#E12:83_1_1_2"><span class="lab"><font color="Green" title="E74">A8</font></span></a>, <a class="txt" href="#E13:83_1_1_2"><span class="lab"><font color="Green" title="E75">A9</font></span></a>, <a class="ref" href="euclidlp.html#T74" target="_self" title="EUCLIDLP:th.74">Th79</a></span>;<br/><a NAME="E16:83_1_1_2"/>
 not <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:83_1_1_2_1"/>
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"> contradiction</span><br/>

<a NAME="E2:83_1_1_2_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">a</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Maroon" title="c5">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">x0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_1_1_2"><span class="lab"><font color="Green" title="E70">A4</font></span></a>, <a class="txt" href="#E12:83_1_1_2"><span class="lab"><font color="Green" title="E74">A8</font></span></a>, <a class="txt" href="#E13:83_1_1_2"><span class="lab"><font color="Green" title="E75">A9</font></span></a>, <a class="ref" href="euclidlp.html#T70" target="_self" title="EUCLIDLP:th.70">Th75</a></span>;<br/>
<a NAME="E3:83_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">x0</font> <a href="euclidlp.html#R2" title="EUCLIDLP:pred.2">//</a> <font color="Maroon" title="c6">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">x0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83_1_1_2"><span class="lab"><font color="Green" title="E73">A7</font></span></a>, <a class="txt" href="#E14:83_1_1_2"><span class="lab"><font color="Green" title="E76">A10</font></span></a>, <a class="ref" href="euclidlp.html#D1" target="_self" title="EUCLIDLP:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:83_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:83_1_1_2"><span class="lab"><font color="Green" title="E77">A11</font></span></a>, <a class="txt" href="euclidlp.html#E45"><span class="lab"><font color="Green" title="E39">Lm3</font></span></a>, <a class="ref" href="euclidlp.html#T45" target="_self" title="EUCLIDLP:th.45">Th50</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/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E17:83_1_1_2"/>
<font color="Maroon" title="c2">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">L2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:83_1_1_2"><span class="lab"><font color="Green" title="E72">A6</font></span></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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:83"/>
<font color="Maroon" title="c2">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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"> verum</span><br/>


</div>
