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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  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><br/> for <font color="Olive" title="b4">P</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="euclidlp.html#K5" title="EUCLIDLP:func.5">plane_of_REAL</a> <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">P</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">P</font> &amp; <font color="Olive" title="b3">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">P</font> &amp; <font color="Olive" title="b2">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Olive" title="b1">x1</font>,<font color="Olive" title="b3">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Olive" title="b1">x1</font> <a href="euclidlp.html#R3" title="EUCLIDLP:pred.3">are_lindependent2</a>  holds <br/><font color="Olive" title="b4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="euclidlp.html#K5" title="EUCLIDLP:func.5">plane_of_REAL</a> <font color="Maroon" title="c1">n</font>  st <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c4">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font> <a href="euclidlp.html#R3" title="EUCLIDLP:pred.3">are_lindependent2</a>  holds <br/><font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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/><span class="kw">let </span><font color="Maroon" title="c5">P</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="euclidlp.html#K5" title="EUCLIDLP:func.5">plane_of_REAL</a> <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c4">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font> <a href="euclidlp.html#R3" title="EUCLIDLP:pred.3">are_lindependent2</a>  implies <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:109"/><span class="lab"><font color="Green" title="E81">A1</font></span>: 
<font color="Maroon" title="c2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <span class="kw">and </span><br/><a NAME="E2:109"/><span class="lab"><font color="Green" title="E82">A2</font></span>: 
<font color="Maroon" title="c3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <span class="kw">and </span><br/><a NAME="E3:109"/><span class="lab"><font color="Green" title="E83">A3</font></span>: 
<font color="Maroon" title="c4">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <span class="kw">and </span><br/><a NAME="E4:109"/><span class="lab"><font color="Green" title="E84">A4</font></span>: 
<font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font> <a href="euclidlp.html#R3" title="EUCLIDLP:pred.3">are_lindependent2</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="E5:109"/>
<font color="Maroon" title="c5">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclidlp.html#K5" title="EUCLIDLP:func.5">plane_of_REAL</a> <font color="Maroon" title="c1">n</font>
 
;<br/>
<a NAME="E6:109"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">P9</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P9</font> &amp;  ex <font color="Olive" title="b2">y1</font>, <font color="Olive" title="b3">y2</font>, <font color="Olive" title="b4">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> st <font color="Olive" title="b1">P9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> <font color="Olive" title="b2">y1</font>,<font color="Olive" title="b3">y2</font>,<font color="Olive" title="b4">y3</font> )
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">y1</font>, <font color="Maroon" title="c7">y2</font>, <font color="Maroon" title="c8">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><span class="kw"> such that </span><br/><a NAME="E8:109"/><span class="lab"><font color="Green" title="E85">A5</font></span>: 
<font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclidlp.html#K4" title="EUCLIDLP:func.4">plane</a> <font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c7">y2</font>,<font color="Maroon" title="c8">y3</font>
 ;<br/>
<a NAME="E9:109"/>
 ex <font color="Olive" title="b1">x9</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> st <br/>( <font color="Maroon" title="c3">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x9</font> &amp;  ex <font color="Olive" title="b2">a19</font>, <font color="Olive" title="b3">a29</font>, <font color="Olive" title="b4">a39</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="b2">a19</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b3">a29</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b4">a39</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b1">x9</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="b2">a19</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">y1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">a29</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">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="b4">a39</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">y3</font></span>)</span> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:109"><span class="lab"><font color="Green" title="E82">A2</font></span></a>, <a class="txt" href="#E8:109"><span class="lab"><font color="Green" title="E85">A5</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">a21</font>, <font color="Maroon" title="c10">a22</font>, <font color="Maroon" title="c11">a23</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E11:109"/><span class="lab"><font color="Green" title="E86">A6</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">a21</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c10">a22</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c11">a23</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">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="c9">a21</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">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">a22</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">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">a23</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">y3</font></span>)</span> )
 ;<br/>
<a NAME="E12:109"/>
 ex <font color="Olive" title="b1">x9</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> st <br/>( <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x9</font> &amp;  ex <font color="Olive" title="b2">a19</font>, <font color="Olive" title="b3">a29</font>, <font color="Olive" title="b4">a39</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="b2">a19</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b3">a29</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b4">a39</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b1">x9</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="b2">a19</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">y1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">a29</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">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="b4">a39</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">y3</font></span>)</span> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:109"><span class="lab"><font color="Green" title="E81">A1</font></span></a>, <a class="txt" href="#E8:109"><span class="lab"><font color="Green" title="E85">A5</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">a11</font>, <font color="Maroon" title="c13">a12</font>, <font color="Maroon" title="c14">a13</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E14:109"/><span class="lab"><font color="Green" title="E87">A7</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c12">a11</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c14">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="c12">a11</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">y1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">a12</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">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="c14">a13</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">y3</font></span>)</span> )
 ;<br/>
<a NAME="E15:109"/>
 ex <font color="Olive" title="b1">x9</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> st <br/>( <font color="Maroon" title="c4">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x9</font> &amp;  ex <font color="Olive" title="b2">a19</font>, <font color="Olive" title="b3">a29</font>, <font color="Olive" title="b4">a39</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="b2">a19</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b3">a29</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b4">a39</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b1">x9</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="b2">a19</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">y1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">a29</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">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="b4">a39</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">y3</font></span>)</span> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109"><span class="lab"><font color="Green" title="E83">A3</font></span></a>, <a class="txt" href="#E8:109"><span class="lab"><font color="Green" title="E85">A5</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">a31</font>, <font color="Maroon" title="c16">a32</font>, <font color="Maroon" title="c17">a33</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E17:109"/><span class="lab"><font color="Green" title="E88">A8</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c15">a31</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c16">a32</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c17">a33</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c4">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="c15">a31</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">y1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">a32</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">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="c17">a33</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">y3</font></span>)</span> )
 ;<br/>
<a NAME="E18:109"/>
<font color="Maroon" title="c4">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">a32</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<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></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">a33</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:109"><span class="lab"><font color="Green" title="E88">A8</font></span></a>, <a class="ref" href="euclidlp.html#T32" target="_self" title="EUCLIDLP:th.32">Th32</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E89">A9</font></span>: <font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">a32</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<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></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">a33</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">y1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">a12</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p5">(<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></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">a13</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:109"><span class="lab"><font color="Green" title="E87">A7</font></span></a>, <a class="ref" href="euclidlp.html#T32" target="_self" title="EUCLIDLP:th.32">Th32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">a32</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<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></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">a33</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</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"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c6">y1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">a12</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p5">(<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></span>)</span></span>)</span></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="c14">a13</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T13" target="_self" title="EUCLIDLP:th.13">Th13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c6">y1</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="c16">a32</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<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></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">a12</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p5">(<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></span>)</span></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="c17">a33</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></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="c14">a13</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T22" target="_self" title="EUCLIDLP:th.22">Th22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</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"><font color="Maroon" title="c16">a32</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<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></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">a12</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p5">(<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></span>)</span></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="c17">a33</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></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="c14">a13</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T7" target="_self" title="EUCLIDLP:th.7">Th7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</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"><font color="Maroon" title="c16">a32</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<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></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="c17">a33</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></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="c14">a13</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T16" target="_self" title="EUCLIDLP:th.16">Th16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</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"><font color="Maroon" title="c16">a32</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<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></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="c17">a33</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T16" target="_self" title="EUCLIDLP:th.16">Th16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">a32</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<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></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="c17">a33</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_4.html#T1" title="EUCLID_4:th.1">EUCLID_4:1</a></span>
;<br/>
<a NAME="E20:109"/><span class="lab"><font color="Green" title="E90">A10</font></span>: 
<font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">a12</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<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></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">a13</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:109"><span class="lab"><font color="Green" title="E87">A7</font></span></a>, <a class="ref" href="euclidlp.html#T32" target="_self" title="EUCLIDLP:th.32">Th32</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">a22</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<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></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a23</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">y1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">a12</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p5">(<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></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">a13</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:109"><span class="lab"><font color="Green" title="E86">A6</font></span></a>, <a class="ref" href="euclidlp.html#T32" target="_self" title="EUCLIDLP:th.32">Th32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">a22</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<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></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a23</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</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"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c6">y1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">a12</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p5">(<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></span>)</span></span>)</span></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="c14">a13</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T13" target="_self" title="EUCLIDLP:th.13">Th13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <font color="Maroon" title="c6">y1</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="c10">a22</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<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></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">a12</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p5">(<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></span>)</span></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="c11">a23</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></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="c14">a13</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T22" target="_self" title="EUCLIDLP:th.22">Th22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</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"><font color="Maroon" title="c10">a22</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<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></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">a12</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p5">(<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></span>)</span></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="c11">a23</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></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="c14">a13</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T7" target="_self" title="EUCLIDLP:th.7">Th7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</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"><font color="Maroon" title="c10">a22</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<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></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="c11">a23</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></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="c14">a13</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T16" target="_self" title="EUCLIDLP:th.16">Th16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</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"><font color="Maroon" title="c10">a22</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<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></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="c11">a23</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T16" target="_self" title="EUCLIDLP:th.16">Th16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">a22</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<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></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="c11">a23</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_4.html#T1" title="EUCLID_4:th.1">EUCLID_4:1</a></span>
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">c1</font>, <font color="Maroon" title="c19">c2</font>, <font color="Maroon" title="c20">d1</font>, <font color="Maroon" title="c21">d2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E23:109"/><span class="lab"><font color="Green" title="E91">A11</font></span>: 
( <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#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">c1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</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">c2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">d1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</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="c21">d2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:109"><span class="lab"><font color="Green" title="E84">A4</font></span></a>, <a class="txt" href="#E19:109"><span class="lab"><font color="Green" title="E89">A9</font></span></a>, <a class="ref" href="euclidlp.html#T41" target="_self" title="EUCLIDLP:th.41">Th41</a></span>;<br/>
<a NAME="E24:109"/><span class="lab"><font color="Green" title="E92">A12</font></span>: 
<font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y1</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="c13">a12</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<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></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">a13</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:109"><span class="lab"><font color="Green" title="E90">A10</font></span></a>, <a class="ref" href="rvsum_1.html#T32" title="RVSUM_1:th.32">RVSUM_1:32</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> implies <font color="Maroon" title="c22">y</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="c3">x2</font>,<font color="Maroon" title="c4">x3</font> )</span><br/><span class="kw">assume </span><a NAME="E1:109_3"/>
<font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c22">y</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="c3">x2</font>,<font color="Maroon" title="c4">x3</font></span><br/><a NAME="E2:109_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x9</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> st <br/>( <font color="Maroon" title="c22">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x9</font> &amp;  ex <font color="Olive" title="b2">a19</font>, <font color="Olive" title="b3">a29</font>, <font color="Olive" title="b4">a39</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="b2">a19</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b3">a29</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b4">a39</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b1">x9</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="b2">a19</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">y1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">a29</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">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="b4">a39</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">y3</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:109"><span class="lab"><font color="Green" title="E85">A5</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">b1</font>, <font color="Maroon" title="c24">b2</font>, <font color="Maroon" title="c25">b3</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:109_3"/><span class="lab"><font color="Green" title="E93">A13</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c23">b1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c24">b2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c25">b3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c22">y</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="c23">b1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">y1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">b2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">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="c25">b3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">y3</font></span>)</span> )
 ;<br/><font color="Maroon" title="c22">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">y1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">b2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<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></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">b3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:109_3"><span class="lab"><font color="Green" title="E93">A13</font></span></a>, <a class="ref" href="euclidlp.html#T32" target="_self" title="EUCLIDLP:th.32">Th32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">a12</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p5">(<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></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">a13</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">b2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<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></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">b3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:109"><span class="lab"><font color="Green" title="E92">A12</font></span></a>, <a class="ref" href="euclidlp.html#T11" target="_self" title="EUCLIDLP:th.11">Th11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">a12</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p5">(<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></span>)</span></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">a13</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">b2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<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></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">b3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T60" title="RVSUM_1:th.60">RVSUM_1:60</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p4">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">a12</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p0">(<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></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">b2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<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></span>)</span></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="c14">a13</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</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="c25">b3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T32" title="RVSUM_1:th.32">RVSUM_1:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">a12</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p0">(<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></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">b2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p5">(<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></span>)</span></span>)</span></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="c14">a13</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</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="c25">b3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T32" title="RVSUM_1:th.32">RVSUM_1:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">a12</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p5">(<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></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">b2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<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></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"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">a13</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c25">b3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T32" title="RVSUM_1:th.32">RVSUM_1:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x1</font> <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">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<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></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"><a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">a13</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c25">b3</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T16" target="_self" title="EUCLIDLP:th.16">Th16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x1</font> <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">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<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></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="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T16" target="_self" title="EUCLIDLP:th.16">Th16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x1</font> <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="c24">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">c1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></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="c24">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">c2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span></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="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">d1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</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="c21">d2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:109"><span class="lab"><font color="Green" title="E91">A11</font></span></a>, <a class="ref" href="euclid_4.html#T6" title="EUCLID_4:th.6">EUCLID_4:6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x1</font> <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="c24">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">c1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></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="c24">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">c2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span></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"><font color="Maroon" title="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">d1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></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="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">d2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_4.html#T6" title="EUCLID_4:th.6">EUCLID_4:6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x1</font> <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="c24">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c18">c1</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</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="c24">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">c2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span></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"><font color="Maroon" title="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">d1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></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="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">d2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_4.html#T4" title="EUCLID_4:th.4">EUCLID_4:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x1</font> <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="c24">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c18">c1</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</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"><span class="p5">(<span class="default"><font color="Maroon" title="c24">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c19">c2</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">d1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></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="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">d2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_4.html#T4" title="EUCLID_4:th.4">EUCLID_4:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x1</font> <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="c24">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c18">c1</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</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"><span class="p5">(<span class="default"><font color="Maroon" title="c24">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c19">c2</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c20">d1</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</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="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">d2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_4.html#T4" title="EUCLID_4:th.4">EUCLID_4:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x1</font> <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="c24">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c18">c1</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</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"><span class="p5">(<span class="default"><font color="Maroon" title="c24">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c19">c2</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c20">d1</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</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="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c21">d2</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_4.html#T4" title="EUCLID_4:th.4">EUCLID_4:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x1</font> <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="c24">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c18">c1</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></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="c24">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c19">c2</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</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="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c20">d1</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</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="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c21">d2</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T32" title="RVSUM_1:th.32">RVSUM_1:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c24">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c18">c1</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></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"><span class="p5">(<span class="default"><font color="Maroon" title="c24">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c19">c2</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></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="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c20">d1</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</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"><font color="Maroon" title="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c21">d2</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T32" title="RVSUM_1:th.32">RVSUM_1:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c24">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c18">c1</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></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"><span class="p5">(<span class="default"><font color="Maroon" title="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c20">d1</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></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="c24">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c19">c2</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</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"><font color="Maroon" title="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c21">d2</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T32" title="RVSUM_1:th.32">RVSUM_1:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x1</font> <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"><span class="p0">(<span class="default"><font color="Maroon" title="c24">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c18">c1</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c20">d1</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span></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="c24">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c19">c2</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</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"><font color="Maroon" title="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c21">d2</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T32" title="RVSUM_1:th.32">RVSUM_1:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x1</font> <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="c24">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c18">c1</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</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"><span class="p5">(<span class="default"><font color="Maroon" title="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c20">d1</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c24">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c19">c2</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</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="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c21">d2</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T32" title="RVSUM_1:th.32">RVSUM_1:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x1</font> <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="c24">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c18">c1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c20">d1</font></span>)</span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</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">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c19">c2</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</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="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c21">d2</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_4.html#T7" title="EUCLID_4:th.7">EUCLID_4:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x1</font> <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="c24">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c18">c1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c20">d1</font></span>)</span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</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">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c19">c2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c21">d2</font></span>)</span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_4.html#T7" title="EUCLID_4:th.7">EUCLID_4:7</a></span>
;<br/><a NAME="E6:109_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c22">y</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">a</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c24">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c18">c1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c20">d1</font></span>)</span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">x2</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">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c19">c2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c21">d2</font></span>)</span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">x3</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <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="c24">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c18">c1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c20">d1</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="c24">b2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">a12</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c19">c2</font></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"><font color="Maroon" title="c25">b3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">a13</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c21">d2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#T33" target="_self" title="EUCLIDLP:th.33">Th33</a></span>;<br/><span class="kw">hence </span><a NAME="E7:109_3"/>
<font color="Maroon" title="c22">y</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="c3">x2</font>,<font color="Maroon" title="c4">x3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E26:109"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A14</font></span>: 
<font color="Maroon" title="c5">P</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="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E27:109"/>
 <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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:109"><span class="lab"><font color="Green" title="E81">A1</font></span></a>, <a class="txt" href="#E2:109"><span class="lab"><font color="Green" title="E82">A2</font></span></a>, <a class="txt" href="#E3:109"><span class="lab"><font color="Green" title="E83">A3</font></span></a>, <a class="txt" href="#E8:109"><span class="lab"><font color="Green" title="E85">A5</font></span></a>, <a class="ref" href="euclidlp.html#T88" target="_self" title="EUCLIDLP:th.88">Th88</a></span>;<br/>
<span class="kw">hence </span><a NAME="E28:109"/>
<font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="#E26:109"><span class="lab"><font color="Green" title="E93">A14</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
