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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">x1</font>, <font color="Maroon" title="c3">y1</font>, <font color="Maroon" title="c4">y2</font>, <font color="Maroon" title="c5">y3</font>, <font color="Maroon" title="c6">x2</font>, <font color="Maroon" title="c7">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>;<br/>













<b>assume </b><a NAME="E1:94"/><i><font color="Green" title="E77">A1</font></i>: 
( <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font>,<font color="Maroon" title="c5">y3</font> &amp; <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font>,<font color="Maroon" title="c5">y3</font> &amp; <font color="Maroon" title="c7">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font>,<font color="Maroon" title="c5">y3</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">x1'</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="E3:94"/><i><font color="Green" title="E78">A2</font></i>: 
( <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x1'</font> &amp;  ex <font color="Olive" title="b1">a11</font>, <font color="Olive" title="b2">a12</font>, <font color="Olive" title="b3">a13</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b1">a11</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b2">a12</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b3">a13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c8">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">a11</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">a12</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">y2</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">a13</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c5">y3</font></span>)</span> ) )
 ;<br/>
<b>consider </b><font color="Maroon" title="c9">a11</font>, <font color="Maroon" title="c10">a12</font>, <font color="Maroon" title="c11">a13</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:94"/><i><font color="Green" title="E79">A3</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">a11</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c10">a12</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c11">a13</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a11</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">a12</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">y2</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a13</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c5">y3</font></span>)</span> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E3:94"><i><font color="Green" title="E78">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">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><b> such that </b><br/><a NAME="E7:94"/><i><font color="Green" title="E80">A4</font></i>: 
( <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x2'</font> &amp;  ex <font color="Olive" title="b1">a21</font>, <font color="Olive" title="b2">a22</font>, <font color="Olive" title="b3">a23</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b1">a21</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b2">a22</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b3">a23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c12">x2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">a21</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">a22</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">y2</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">a23</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c5">y3</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:94"><i><font color="Green" title="E77">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">a21</font>, <font color="Maroon" title="c14">a22</font>, <font color="Maroon" title="c15">a23</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E9:94"/><i><font color="Green" title="E81">A5</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c13">a21</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c14">a22</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">a23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">a21</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">a22</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">y2</font></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">a23</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c5">y3</font></span>)</span> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E7:94"><i><font color="Green" title="E80">A4</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">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="E11:94"/><i><font color="Green" title="E82">A6</font></i>: 
( <font color="Maroon" title="c7">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x3'</font> &amp;  ex <font color="Olive" title="b1">a31</font>, <font color="Olive" title="b2">a32</font>, <font color="Olive" title="b3">a33</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b1">a31</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b2">a32</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b3">a33</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c16">x3'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">a31</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">a32</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">y2</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">a33</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c5">y3</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:94"><i><font color="Green" title="E77">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">a31</font>, <font color="Maroon" title="c18">a32</font>, <font color="Maroon" title="c19">a33</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E13:94"/><i><font color="Green" title="E83">A7</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c17">a31</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c18">a32</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c19">a33</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c7">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">a31</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">a32</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">y2</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">a33</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c5">y3</font></span>)</span> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E11:94"><i><font color="Green" title="E82">A6</font></i></a></i>;<br/>
<b>let </b><font color="Maroon" title="c20">x</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><br/>

<b>assume </b><a NAME="E14:94"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN: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="c6">x2</font>,<font color="Maroon" title="c7">x3</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c21">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><b> such that </b><br/><a NAME="E16:94"/><i><font color="Green" title="E84">A8</font></i>: 
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">x'</font> &amp;  ex <font color="Olive" title="b1">b1</font>, <font color="Olive" title="b2">b2</font>, <font color="Olive" title="b3">b3</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b1">b1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b2">b2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b3">b3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c21">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">b1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">b2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">x2</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">b3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">x3</font></span>)</span> ) )
 ;<br/>
<b>consider </b><font color="Maroon" title="c22">b1</font>, <font color="Maroon" title="c23">b2</font>, <font color="Maroon" title="c24">b3</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E18:94"/><i><font color="Green" title="E85">A9</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c22">b1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c23">b2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c24">b3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">b1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">b2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">x2</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">b3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">x3</font></span>)</span> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E16:94"><i><font color="Green" title="E84">A8</font></i></a></i>;<br/>
<i><font color="Green" title="E86">A10</font></i>: <font color="Maroon" title="c20">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c22">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c9">a11</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c22">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">a12</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">y2</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c22">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c11">a13</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c5">y3</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">b2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c13">a21</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c14">a22</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">y2</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">a23</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c5">y3</font></span>)</span></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="c24">b3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c17">a31</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">a32</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">y2</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">a33</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c5">y3</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E5:94"><i><font color="Green" title="E79">A3</font></i></a>, <a class="txt" href="euclidlp.html#E9:94"><i><font color="Green" title="E81">A5</font></i></a>, <a class="txt" href="euclidlp.html#E13:94"><i><font color="Green" title="E83">A7</font></i></a>, <a class="txt" href="euclidlp.html#E18:94"><i><font color="Green" title="E85">A9</font></i></a>, <a class="ref" href="euclidlp.html#T27" target="_self" title="EUCLIDLP:th.27">Th27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c22">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c9">a11</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c22">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">a12</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">y2</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c22">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c11">a13</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c5">y3</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c23">b2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c13">a21</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c23">b2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c14">a22</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">y2</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c23">b2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c15">a23</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c5">y3</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="c24">b3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c17">a31</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">a32</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">y2</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">a33</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c5">y3</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T27" target="_self" title="EUCLIDLP:th.27">Th27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c22">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c9">a11</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c22">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">a12</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">y2</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c22">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c11">a13</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c5">y3</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c23">b2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c13">a21</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c23">b2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c14">a22</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">y2</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c23">b2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c15">a23</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c5">y3</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c24">b3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c17">a31</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c24">b3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c18">a32</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">y2</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">b3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c19">a33</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c5">y3</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T27" target="_self" title="EUCLIDLP:th.27">Th27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c22">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c9">a11</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c23">b2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c13">a21</font></span>)</span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c22">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">a12</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c23">b2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c14">a22</font></span>)</span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">y2</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c22">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c11">a13</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c23">b2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c15">a23</font></span>)</span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c5">y3</font></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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c24">b3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c17">a31</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c24">b3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c18">a32</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">y2</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">b3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c19">a33</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c5">y3</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T29" target="_self" title="EUCLIDLP:th.29">Th29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c22">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c9">a11</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c23">b2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c13">a21</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">b3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c17">a31</font></span>)</span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">y1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c22">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">a12</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c23">b2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c14">a22</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">b3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c18">a32</font></span>)</span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">y2</font></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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c22">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c11">a13</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c23">b2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c15">a23</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">b3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c19">a33</font></span>)</span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c5">y3</font></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T29" target="_self" title="EUCLIDLP:th.29">Th29</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c22">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c9">a11</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c23">b2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c13">a21</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">b3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c17">a31</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c22">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">a12</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c23">b2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c14">a22</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">b3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c18">a32</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1: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="c22">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c11">a13</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">b2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c15">a23</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">b3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c19">a33</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">b1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">a11</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c10">a12</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c11">a13</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">b2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">a21</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c14">a22</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c15">a23</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">b3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">a31</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c18">a32</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c19">a33</font></span>)</span></span>)</span>

<br/>.= 
1
<b>by </b><i><a class="txt" href="euclidlp.html#E5:94"><i><font color="Green" title="E79">A3</font></i></a>, <a class="txt" href="euclidlp.html#E9:94"><i><font color="Green" title="E81">A5</font></i></a>, <a class="txt" href="euclidlp.html#E13:94"><i><font color="Green" title="E83">A7</font></i></a>, <a class="txt" href="euclidlp.html#E18:94"><i><font color="Green" title="E85">A9</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E21:94"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font>,<font color="Maroon" title="c5">y3</font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E19:94"><i><font color="Green" title="E86">A10</font></i></a></i>;<br/>


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