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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">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> 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="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> (<font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font>) &amp;  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> (<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> (<font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font>) &amp;  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> (<font color="Olive" title="b3">x3</font>,<font color="Olive" title="b1">x1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> (<font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">x2</font>, <font color="Maroon" title="c4">x3</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> (<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">x3</font>) &amp;  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> (<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">x3</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> (<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">x3</font>) &amp;  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> (<font color="Maroon" title="c4">x3</font>,<font color="Maroon" title="c2">x1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> (<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">x3</font>) )</span><br/>







<a NAME="E1:106"/><span class="lab"><font color="Green" title="E76">A1</font></span>: 
<font color="Maroon" title="c4">x3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> (<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">x3</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T82" target="_self" onmouseover="rs('euclidlp/T82')" onmouseout="rh()">Th82</a></span>;<br/>
<a NAME="E2:106"/>
( <font color="Maroon" title="c2">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> (<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">x3</font>) &amp; <font color="Maroon" title="c3">x2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> (<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">x3</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T82" target="_self" onmouseover="rs('euclidlp/T82')" onmouseout="rh()">Th82</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:106"/>
(  <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>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> (<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">x3</font>) &amp;  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> (<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">x3</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> (<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">x3</font>) &amp;  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> (<font color="Maroon" title="c4">x3</font>,<font color="Maroon" title="c2">x1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> (<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">x3</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106"><span class="lab"><font color="Green" title="E76">A1</font></span></a>, <a class="ref" href="euclidlp.html#T85" target="_self" onmouseover="rs('euclidlp/T85')" onmouseout="rh()">Th85</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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