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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">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#R5" title="EUCLIDLP:pred.5">//</a> <font color="Olive" title="b2">L2</font> &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/><font color="Olive" title="b1">L1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">L2</font></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">L1</font> <a href="euclidlp.html#R5" title="EUCLIDLP:pred.5">//</a> <font color="Maroon" title="c3">L2</font> &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 <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> )</span><br/>





<b>assume </b><a NAME="E1:79"/><i><font color="Green" title="E66">A1</font></i>: 
( <font color="Maroon" title="c2">L1</font> <a href="euclidlp.html#R5" title="EUCLIDLP:pred.5">//</a> <font color="Maroon" title="c3">L2</font> &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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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></span><br/>

<b>assume </b><a NAME="E2:79"/><i><font color="Green" title="E67">A2</font></i>: 
not <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>consider </b><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">x2</font>, <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="E4:79"/><i><font color="Green" title="E68">A3</font></i>: 
( <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> &amp; <font color="Maroon" title="c3">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="c6">y1</font>,<font color="Maroon" title="c7">y2</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">x1</font> <a href="euclidlp.html#R2" title="EUCLIDLP:pred.2">//</a> <font color="Maroon" title="c7">y2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:79"><i><font color="Green" title="E66">A1</font></i></a>, <a class="ref" href="euclidlp.html#D7" target="_self" title="EUCLIDLP:def.7">Def7</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:79"/><i><font color="Green" title="E69">A4</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L1</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L2</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E2:79"><i><font color="Green" title="E67">A2</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c8">x</font> as    <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>by </b><i><a class="txt" href="euclidlp.html#E6:79"><i><font color="Green" title="E69">A4</font></i></a></i>;<br/>
<a NAME="E8:79"/><i><font color="Green" title="E70">A5</font></i>: 
( <font color="Maroon" title="c5">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">x1</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">y2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</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#E4:79"><i><font color="Green" title="E68">A3</font></i></a>, <a class="ref" href="euclidlp.html#D1" target="_self" title="EUCLIDLP:def.1">Def1</a></i>;<br/>
<a NAME="E9:79"/><b>then </b>
( <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> &amp; <font color="Maroon" title="c6">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">y2</font> )
 
<b>by </b><i><a class="ref" href="euclidlp.html#T14" target="_self" title="EUCLIDLP:th.14">Th14</a></i>;<br/>
<a NAME="E10:79"/><b>then </b><i><font color="Green" title="E71">A6</font></i>: 
( <font color="Maroon" title="c2">L1</font> <a href="euclid_4.html#NR1" title="EUCLID_4:NR.1">is_line</a>  &amp; <font color="Maroon" title="c3">L2</font> <a href="euclid_4.html#NR1" title="EUCLID_4:NR.1">is_line</a>  )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E4:79"><i><font color="Green" title="E68">A3</font></i></a>, <a class="ref" href="euclid_4.html#D2" title="EUCLID_4:def.2">EUCLID_4:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">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="E12:79"/><i><font color="Green" title="E72">A7</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">x0</font> &amp; <font color="Maroon" title="c10">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L1</font> )
 <b>by </b><i><a class="ref" href="euclid_4.html#T15" title="EUCLID_4:th.15">EUCLID_4:15</a></i>;<br/>
<a NAME="E13:79"/><i><font color="Green" title="E73">A8</font></i>: 
<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="c9">x</font>,<font color="Maroon" title="c10">x0</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E6:79"><i><font color="Green" title="E69">A4</font></i></a>, <a class="txt" href="euclidlp.html#E10:79"><i><font color="Green" title="E71">A6</font></i></a>, <a class="txt" href="euclidlp.html#E12:79"><i><font color="Green" title="E72">A7</font></i></a>, <a class="ref" href="euclidlp.html#T35" target="_self" title="EUCLIDLP:th.35">Th35</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">y0</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="E15:79"/><i><font color="Green" title="E74">A9</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">y0</font> &amp; <font color="Maroon" title="c11">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L2</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E10:79"><i><font color="Green" title="E71">A6</font></i></a>, <a class="ref" href="euclid_4.html#T15" title="EUCLID_4:th.15">EUCLID_4:15</a></i>;<br/>
<a NAME="E16:79"/><i><font color="Green" title="E75">A10</font></i>: 
<font color="Maroon" title="c3">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="c9">x</font>,<font color="Maroon" title="c11">y0</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E6:79"><i><font color="Green" title="E69">A4</font></i></a>, <a class="txt" href="euclidlp.html#E10:79"><i><font color="Green" title="E71">A6</font></i></a>, <a class="txt" href="euclidlp.html#E15:79"><i><font color="Green" title="E74">A9</font></i></a>, <a class="ref" href="euclidlp.html#T35" target="_self" title="EUCLIDLP:th.35">Th35</a></i>;<br/>
<a NAME="E17:79"/><i><font color="Green" title="E76">A11</font></i>: 
( <font color="Maroon" title="c10">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">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="c11">y0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">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#E12:79"><i><font color="Green" title="E72">A7</font></i></a>, <a class="txt" href="euclidlp.html#E15:79"><i><font color="Green" title="E74">A9</font></i></a>, <a class="ref" href="euclidlp.html#T14" target="_self" title="EUCLIDLP:th.14">Th14</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">a</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E19:79"/><i><font color="Green" title="E77">A12</font></i>: 
<font color="Maroon" title="c10">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">x1</font></span>)</span>
 <b>by </b><i><a class="txt" href="euclidlp.html#E4:79"><i><font color="Green" title="E68">A3</font></i></a>, <a class="txt" href="euclidlp.html#E6:79"><i><font color="Green" title="E69">A4</font></i></a>, <a class="txt" href="euclidlp.html#E12:79"><i><font color="Green" title="E72">A7</font></i></a>, <a class="ref" href="euclidlp.html#T36" target="_self" title="EUCLIDLP:th.36">Th36</a></i>;<br/>
<a NAME="E20:79"/><i><font color="Green" title="E78">A13</font></i>: 
<font color="Maroon" title="c10">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">x</font> <a href="euclidlp.html#R2" title="EUCLIDLP:pred.2">//</a> <font color="Maroon" title="c5">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c4">x1</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E8:79"><i><font color="Green" title="E70">A5</font></i></a>, <a class="txt" href="euclidlp.html#E17:79"><i><font color="Green" title="E76">A11</font></i></a>, <a class="txt" href="euclidlp.html#E19:79"><i><font color="Green" title="E77">A12</font></i></a>, <a class="ref" href="euclidlp.html#D1" target="_self" title="EUCLIDLP:def.1">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">b</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E22:79"/><i><font color="Green" title="E79">A14</font></i>: 
<font color="Maroon" title="c11">y0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span>
 <b>by </b><i><a class="txt" href="euclidlp.html#E4:79"><i><font color="Green" title="E68">A3</font></i></a>, <a class="txt" href="euclidlp.html#E6:79"><i><font color="Green" title="E69">A4</font></i></a>, <a class="txt" href="euclidlp.html#E15:79"><i><font color="Green" title="E74">A9</font></i></a>, <a class="ref" href="euclidlp.html#T36" target="_self" title="EUCLIDLP:th.36">Th36</a></i>;<br/>
<a NAME="E23:79"/><i><font color="Green" title="E80">A15</font></i>: 
<font color="Maroon" title="c11">y0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">x</font> <a href="euclidlp.html#R2" title="EUCLIDLP:pred.2">//</a> <font color="Maroon" title="c7">y2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E8:79"><i><font color="Green" title="E70">A5</font></i></a>, <a class="txt" href="euclidlp.html#E17:79"><i><font color="Green" title="E76">A11</font></i></a>, <a class="txt" href="euclidlp.html#E22:79"><i><font color="Green" title="E79">A14</font></i></a>, <a class="ref" href="euclidlp.html#D1" target="_self" title="EUCLIDLP:def.1">Def1</a></i>;<br/>
<a NAME="E24:79"/>
<font color="Maroon" title="c10">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">x</font> <a href="euclidlp.html#R2" title="EUCLIDLP:pred.2">//</a> <font color="Maroon" title="c7">y2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E4:79"><i><font color="Green" title="E68">A3</font></i></a>, <a class="txt" href="euclidlp.html#E20:79"><i><font color="Green" title="E78">A13</font></i></a>, <a class="ref" href="euclidlp.html#T38" target="_self" title="EUCLIDLP:th.38">Th38</a></i>;<br/>
<a NAME="E25:79"/><b>then </b><i><font color="Green" title="E81">A16</font></i>: 
<font color="Maroon" title="c10">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">x</font> <a href="euclidlp.html#R2" title="EUCLIDLP:pred.2">//</a> <font color="Maroon" title="c11">y0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">x</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E23:79"><i><font color="Green" title="E80">A15</font></i></a>, <a class="ref" href="euclidlp.html#T38" target="_self" title="EUCLIDLP:th.38">Th38</a></i>;<br/>
<a NAME="E26:79"/><i><font color="Green" title="E82">A17</font></i>: 
 <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">x0</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="c9">x</font>,<font color="Maroon" title="c11">y0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c14">y</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">x</font>,<font color="Maroon" title="c10">x0</font> or <font color="Maroon" title="c14">y</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">x</font>,<font color="Maroon" title="c11">y0</font> )</span><br/>

<b>assume </b><a NAME="E1:79_1"/>
<font color="Maroon" title="c14">y</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">x</font>,<font color="Maroon" title="c10">x0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">y</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">x</font>,<font color="Maroon" title="c11">y0</font></span><br/>

<b>then consider </b><font color="Maroon" title="c15">t</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:79_1"/><i><font color="Green" title="E83">A18</font></i>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">t</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">t</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c10">x0</font></span>)</span>
 ;<br/>
<i><font color="Green" title="E84">A19</font></i>: <font color="Maroon" title="c14">y</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">t</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">t</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c10">x0</font></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E3:79_1"><i><font color="Green" title="E83">A18</font></i></a>, <a class="ref" href="euclidlp.html#T16" target="_self" title="EUCLIDLP:th.16">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">t</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">t</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c10">x0</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="c9">x</font> <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="c15">t</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c10">x0</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">t</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c9">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/>.= 
<font color="Maroon" title="c9">x</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">t</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T17" target="_self" title="EUCLIDLP:th.17">Th17</a></i>
;<br/>
<b>consider </b><font color="Maroon" title="c16">a</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E6:79_1"/><i><font color="Green" title="E85">A20</font></i>: 
<font color="Maroon" title="c10">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">y0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="euclidlp.html#E25:79"><i><font color="Green" title="E81">A16</font></i></a>, <a class="ref" href="euclidlp.html#D1" target="_self" title="EUCLIDLP:def.1">Def1</a></i>;<br/>
<font color="Maroon" title="c14">y</font> = 
<font color="Maroon" title="c9">x</font> <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="c15">t</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">y0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E4:79_1"><i><font color="Green" title="E84">A19</font></i></a>, <a class="txt" href="euclidlp.html#E6:79_1"><i><font color="Green" title="E85">A20</font></i></a>, <a class="ref" href="euclid_4.html#T4" title="EUCLID_4:th.4">EUCLID_4:4</a></i>
<br/>.= 
<font color="Maroon" title="c9">x</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">t</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c11">y0</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c15">t</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T17" target="_self" title="EUCLIDLP:th.17">Th17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c15">t</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span></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="c15">t</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c11">y0</font></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">1 <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c15">t</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span></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="c15">t</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c11">y0</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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">t</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c9">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="c15">t</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c11">y0</font></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T16" target="_self" title="EUCLIDLP:th.16">Th16</a></i>
;<br/>
<b>hence </b><a NAME="E8:79_1"/>
<font color="Maroon" title="c14">y</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">x</font>,<font color="Maroon" title="c11">y0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E27:79"/>
 <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">y0</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="c9">x</font>,<font color="Maroon" title="c10">x0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c14">y</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">x</font>,<font color="Maroon" title="c11">y0</font> or <font color="Maroon" title="c14">y</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">x</font>,<font color="Maroon" title="c10">x0</font> )</span><br/>

<b>assume </b><a NAME="E1:79_2"/>
<font color="Maroon" title="c14">y</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">x</font>,<font color="Maroon" title="c11">y0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">y</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">x</font>,<font color="Maroon" title="c10">x0</font></span><br/>

<b>then consider </b><font color="Maroon" title="c15">t</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:79_2"/><i><font color="Green" title="E83">A21</font></i>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">t</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">t</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c11">y0</font></span>)</span>
 ;<br/>
<i><font color="Green" title="E84">A22</font></i>: <font color="Maroon" title="c14">y</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">t</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">t</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c11">y0</font></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E3:79_2"><i><font color="Green" title="E83">A21</font></i></a>, <a class="ref" href="euclidlp.html#T16" target="_self" title="EUCLIDLP:th.16">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">t</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">t</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c11">y0</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="c9">x</font> <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="c15">t</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c11">y0</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">t</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c9">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/>.= 
<font color="Maroon" title="c9">x</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">t</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">y0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T17" target="_self" title="EUCLIDLP:th.17">Th17</a></i>
;<br/>
<b>consider </b><font color="Maroon" title="c16">a</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E6:79_2"/><i><font color="Green" title="E85">A23</font></i>: 
<font color="Maroon" title="c11">y0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="euclidlp.html#E25:79"><i><font color="Green" title="E81">A16</font></i></a>, <a class="ref" href="euclidlp.html#D1" target="_self" title="EUCLIDLP:def.1">Def1</a></i>;<br/>
<font color="Maroon" title="c14">y</font> = 
<font color="Maroon" title="c9">x</font> <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="c15">t</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">x0</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E4:79_2"><i><font color="Green" title="E84">A22</font></i></a>, <a class="txt" href="euclidlp.html#E6:79_2"><i><font color="Green" title="E85">A23</font></i></a>, <a class="ref" href="euclid_4.html#T4" title="EUCLID_4:th.4">EUCLID_4:4</a></i>
<br/>.= 
<font color="Maroon" title="c9">x</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">t</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c10">x0</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c15">t</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T17" target="_self" title="EUCLIDLP:th.17">Th17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c15">t</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span></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="c15">t</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c10">x0</font></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">1 <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c15">t</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span></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="c15">t</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c10">x0</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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">t</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c9">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="c15">t</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c10">x0</font></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T16" target="_self" title="EUCLIDLP:th.16">Th16</a></i>
;<br/>
<b>hence </b><a NAME="E8:79_2"/>
<font color="Maroon" title="c14">y</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">x</font>,<font color="Maroon" title="c10">x0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E28:79"/>
contradiction
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:79"><i><font color="Green" title="E66">A1</font></i></a>, <a class="txt" href="euclidlp.html#E13:79"><i><font color="Green" title="E73">A8</font></i></a>, <a class="txt" href="euclidlp.html#E16:79"><i><font color="Green" title="E75">A10</font></i></a>, <a class="txt" href="euclidlp.html#E26:79"><i><font color="Green" title="E82">A17</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
