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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">x1</font>, <font color="Olive" title="b2">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><br/> for <font color="Olive" title="b3">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>  st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">L</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">L</font> &amp; <font color="Olive" title="b1">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">x2</font> holds <br/>(  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">L</font> &amp; <font color="Olive" title="b3">L</font> is <a href="euclid_4.html#V1" title="EUCLID_4:attr.1">being_line</a> )</span><br/><b>let </b><font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">x2</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>  st <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">L</font> &amp; <font color="Maroon" title="c3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">L</font> &amp; <font color="Maroon" title="c2">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x2</font> holds <br/>(  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> )</span><br/><b>let </b><font color="Maroon" title="c4">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"> ( <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">L</font> &amp; <font color="Maroon" title="c3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">L</font> &amp; <font color="Maroon" title="c2">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x2</font> implies (  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">L</font> &amp; <font color="Maroon" title="c4">L</font> is <a href="euclid_4.html#V1" title="EUCLID_4:attr.1">being_line</a> ) )</span><br/>







<b>assume </b><a NAME="E1:72"/><i><font color="Green" title="E61">A1</font></i>: 
( <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">L</font> &amp; <font color="Maroon" title="c3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">L</font> &amp; <font color="Maroon" title="c2">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">L</font> &amp; <font color="Maroon" title="c4">L</font> is <a href="euclid_4.html#V1" title="EUCLID_4:attr.1">being_line</a> )</span><br/>

<a NAME="E2:72"/>
<font color="Maroon" title="c4">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclidlp.html#K1" title="EUCLIDLP:func.1">line_of_REAL</a> <font color="Maroon" title="c1">n</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c5">L0</font> being   <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><b> such that </b><br/><a NAME="E4:72"/><i><font color="Green" title="E62">A2</font></i>: 
( <font color="Maroon" title="c4">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L0</font> &amp;  ex <font color="Olive" title="b1">y1</font>, <font color="Olive" title="b2">y2</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="Maroon" title="c5">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="Olive" title="b1">y1</font>,<font color="Olive" title="b2">y2</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c6">y1</font>, <font color="Maroon" title="c7">y2</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="E6:72"/><i><font color="Green" title="E63">A3</font></i>: 
<font color="Maroon" title="c4">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="c6">y1</font>,<font color="Maroon" title="c7">y2</font>
 <b>by </b><i><a class="txt" href="#E4:72"><i><font color="Green" title="E62">A2</font></i></a></i>;<br/>
<a NAME="E7:72"/><i><font color="Green" title="E64">A4</font></i>: 
<font color="Maroon" title="c4">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font>
 
<b>by </b><i><a class="txt" href="#E1:72"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="#E6:72"><i><font color="Green" title="E63">A3</font></i></a>, <a class="ref" href="euclid_4.html#T12" title="EUCLID_4:th.12">EUCLID_4:12</a></i>;<br/>
<a NAME="E8:72"/>
 <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">L</font>
 
<b>by </b><i><a class="txt" href="#E1:72"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="euclidlp.html#T53" target="_self" title="EUCLIDLP:th.53">Th53</a></i>;<br/>
<a NAME="E9:72"/><b>then </b>
 <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">L</font>
 
<b>by </b><i><a class="txt" href="#E7:72"><i><font color="Green" title="E64">A4</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E10:72"/>
(  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">L</font> &amp; <font color="Maroon" title="c4">L</font> is <a href="euclid_4.html#V1" title="EUCLID_4:attr.1">being_line</a> )
 <b>by </b><i><a class="txt" href="#E1:72"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="euclid_4.html#D2" title="EUCLID_4:def.2">EUCLID_4:def 2</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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