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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="Red">::  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> is  <a href="euclid_4.html#V1" title="EUCLID_4:attr.1">being_line</a>  &amp; <font color="Olive" title="b2">L2</font> is  <a href="euclid_4.html#V1" title="EUCLID_4:attr.1">being_line</a>  &amp; <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> holds <br/> ex <font color="Olive" title="b3">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 <br/>( <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">L1</font> &amp; not <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">L1</font> is  <a href="euclid_4.html#V1" title="EUCLID_4:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">L2</font> is  <a href="euclid_4.html#V1" title="EUCLID_4:attr.1">being_line</a>  &amp; <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> implies  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 <br/>( <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> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L2</font> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:92"/><span class="lab"><font color="Green" title="E71">A1</font></span>: 
<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">and </span><br/><a NAME="E2:92"/><span class="lab"><font color="Green" title="E72">A2</font></span>: 
<font color="Maroon" title="c3">L2</font> is  <a href="euclid_4.html#V1" title="EUCLID_4:attr.1">being_line</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <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> or  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 <br/>( <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> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L2</font> ) )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">x1</font>, <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="E4:92"/><span class="lab"><font color="Green" title="E73">A3</font></span>: 
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x2</font>
 <span class="kw">and </span><br/><a NAME="E5:92"/><span class="lab"><font color="Green" title="E74">A4</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="#E1:92"><span class="lab"><font color="Green" title="E71">A1</font></span></a>, <a class="ref" href="euclid_4.html#D2" title="EUCLID_4:def.2">EUCLID_4:def 2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:92"/><span class="lab"><font color="Green" title="E75">A5</font></span>: 
<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="Red">::  thesis: </font></span></a><span class="hide">  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 <br/>( <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> &amp; not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L2</font> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_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="E1:92_1_1"/>
(  not <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> or  not <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92"><span class="lab"><font color="Green" title="E72">A2</font></span></a>, <a class="txt" href="#E4:92"><span class="lab"><font color="Green" title="E73">A3</font></span></a>, <a class="txt" href="#E5:92"><span class="lab"><font color="Green" title="E74">A4</font></span></a>, <a class="txt" href="#E6:92"><span class="lab"><font color="Green" title="E75">A5</font></span></a>, <a class="ref" href="euclidlp.html#T35" target="_self" title="EUCLIDLP:th.35">Th35</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1"><span class="kw">case </span></a><a NAME="E1:92_1_1_1"/><span class="lab"><font color="Green" title="E76">A6</font></span>: 
not <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>
 ; <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="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L1</font> &amp; not <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><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c4">x1</font>;<br/><span class="kw">thus </span><a NAME="E2:92_1_1_1"/>
( <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> &amp; not <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="#E5:92"><span class="lab"><font color="Green" title="E74">A4</font></span></a>, <a class="txt" href="#E1:92_1_1_1"><span class="lab"><font color="Green" title="E76">A6</font></span></a>, <a class="ref" href="euclid_4.html#T10" title="EUCLID_4:th.10">EUCLID_4:10</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_2"><span class="kw">case </span></a><a NAME="E1:92_1_1_2"/><span class="lab"><font color="Green" title="E76">A7</font></span>: 
not <font color="Maroon" title="c5">x2</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="Red">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; not <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L2</font> )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x2</font>;<br/><span class="kw">thus </span><a NAME="E2:92_1_1_2"/>
( <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> &amp; not <font color="Maroon" title="c5">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="txt" href="#E5:92"><span class="lab"><font color="Green" title="E74">A4</font></span></a>, <a class="txt" href="#E1:92_1_1_2"><span class="lab"><font color="Green" title="E76">A7</font></span></a>, <a class="ref" href="euclid_4.html#T10" title="EUCLID_4:th.10">EUCLID_4:10</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:92"/>
 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 <br/>( <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> &amp; not <font color="Olive" title="b1">x</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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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