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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> ;<br/><b>let </b><font color="Maroon" title="c2">L0</font>, <font color="Maroon" title="c3">L1</font>, <font color="Maroon" title="c4">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>;<br/>







<b>assume </b><a NAME="E1:70"/><i><font color="Green" title="E64">A1</font></i>: 
( <font color="Maroon" title="c2">L0</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c3">L1</font> &amp; <font color="Maroon" title="c3">L1</font> <a href="euclidlp.html#R5" title="EUCLIDLP:pred.5">//</a> <font color="Maroon" title="c4">L2</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">x0</font>, <font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">x2</font>, <font color="Maroon" title="c8">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><b> such that </b><br/><a NAME="E3:70"/><i><font color="Green" title="E65">A2</font></i>: 
( <font color="Maroon" title="c2">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="c5">x0</font>,<font color="Maroon" title="c6">x1</font> &amp; <font color="Maroon" title="c3">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="c7">x2</font>,<font color="Maroon" title="c8">x3</font> &amp; <font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c5">x0</font> <a href="euclidlp.html#R4" title="EUCLIDLP:pred.4">_|_</a> <font color="Maroon" title="c8">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#D8" target="_self" title="EUCLIDLP:def.8">Def8</a></i>;<br/>
<a NAME="E4:70"/><i><font color="Green" title="E66">A3</font></i>: 
( <font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c5">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> &amp; <font color="Maroon" title="c8">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</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#E3:70"><i><font color="Green" title="E65">A2</font></i></a>, <a class="ref" href="euclidlp.html#D3" target="_self" title="EUCLIDLP:def.3">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">y0</font>, <font color="Maroon" title="c10">y1</font>, <font color="Maroon" title="c11">y2</font>, <font color="Maroon" title="c12">y3</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="E6:70"/><i><font color="Green" title="E67">A4</font></i>: 
( <font color="Maroon" title="c3">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="c9">y0</font>,<font color="Maroon" title="c10">y1</font> &amp; <font color="Maroon" title="c4">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="c11">y2</font>,<font color="Maroon" title="c12">y3</font> &amp; <font color="Maroon" title="c10">y1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">y0</font> <a href="euclidlp.html#R2" title="EUCLIDLP:pred.2">//</a> <font color="Maroon" title="c12">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c11">y2</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:70"><i><font color="Green" title="E64">A1</font></i></a>, <a class="ref" href="euclidlp.html#D7" target="_self" title="EUCLIDLP:def.7">Def7</a></i>;<br/>
<a NAME="E7:70"/><i><font color="Green" title="E68">A5</font></i>: 
( <font color="Maroon" title="c10">y1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">y0</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="c12">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c11">y2</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#E6:70"><i><font color="Green" title="E67">A4</font></i></a>, <a class="ref" href="euclidlp.html#D1" target="_self" title="EUCLIDLP:def.1">Def1</a></i>;<br/>
<a NAME="E8:70"/>
( <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c9">y0</font>,<font color="Maroon" title="c10">y1</font> &amp; <font color="Maroon" title="c8">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c9">y0</font>,<font color="Maroon" title="c10">y1</font> )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E3:70"><i><font color="Green" title="E65">A2</font></i></a>, <a class="txt" href="euclidlp.html#E6:70"><i><font color="Green" title="E67">A4</font></i></a>, <a class="ref" href="euclid_4.html#T10" title="EUCLID_4:th.10">EUCLID_4:10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">a</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E10:70"/><i><font color="Green" title="E69">A6</font></i>: 
<font color="Maroon" title="c8">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">y0</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclidlp.html#T36" target="_self" title="EUCLIDLP:th.36">Th36</a></i>;<br/>
<a NAME="E11:70"/>
<font color="Maroon" title="c8">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font> <a href="euclidlp.html#R2" title="EUCLIDLP:pred.2">//</a> <font color="Maroon" title="c10">y1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">y0</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E10:70"><i><font color="Green" title="E69">A6</font></i></a>, <a class="txt" href="euclidlp.html#E7:70"><i><font color="Green" title="E68">A5</font></i></a>, <a class="txt" href="euclidlp.html#E4:70"><i><font color="Green" title="E66">A3</font></i></a>, <a class="ref" href="euclidlp.html#D1" target="_self" title="EUCLIDLP:def.1">Def1</a></i>;<br/>
<a NAME="E12:70"/><b>then </b>
<font color="Maroon" title="c8">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">x2</font> <a href="euclidlp.html#R2" title="EUCLIDLP:pred.2">//</a> <font color="Maroon" title="c12">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c11">y2</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E6:70"><i><font color="Green" title="E67">A4</font></i></a>, <a class="ref" href="euclidlp.html#T38" target="_self" title="EUCLIDLP:th.38">Th38</a></i>;<br/>
<a NAME="E13:70"/><b>then </b>
<font color="Maroon" title="c6">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c5">x0</font> <a href="euclidlp.html#R4" title="EUCLIDLP:pred.4">_|_</a> <font color="Maroon" title="c12">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c11">y2</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E3:70"><i><font color="Green" title="E65">A2</font></i></a>, <a class="ref" href="euclidlp.html#T49" target="_self" title="EUCLIDLP:th.49">Th49</a></i>;<br/>
<b>hence </b><a NAME="E14:70"/>
<font color="Maroon" title="c2">L0</font> <a href="euclidlp.html#R6" title="EUCLIDLP:pred.6">_|_</a> <font color="Maroon" title="c4">L2</font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E3:70"><i><font color="Green" title="E65">A2</font></i></a>, <a class="txt" href="euclidlp.html#E6:70"><i><font color="Green" title="E67">A4</font></i></a>, <a class="ref" href="euclidlp.html#D8" target="_self" title="EUCLIDLP:def.8">Def8</a></i>;<br/>


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